Mathematics › Applied Mathematics

Optimal Transport in Geometry and Analysis

Works Count: 66967

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Description

This cluster of papers explores the mathematical theory and applications of optimal transport, including topics such as Ricci curvature, Wasserstein distance, metric measure spaces, gradient flows, Sobolev inequalities, the Monge-Kantorovich problem, mean curvature flow, and minimal surfaces.

Keywords

Optimal Transport; Ricci Curvature; Wasserstein Distance; Metric Measure Spaces; Gradient Flows; Sobolev Inequalities; Monge-Kantorovich Problem; Mean Curvature Flow; Geometric Applications; Minimal Surfaces

Metric Structures for Riemannian and Non-Riemannian Spaces

2007-01-01

Mikhael Gromov

B. Path metric spaces C. Examples of path metric spaces D. Arc-wise isometrics 2 Degree and Dilatation A. Topological review B. Elementary properties of dilatations for spheres C. Homotopy counting … B. Path metric spaces C. Examples of path metric spaces D. Arc-wise isometrics 2 Degree and Dilatation A. Topological review B. Elementary properties of dilatations for spheres C. Homotopy counting Lipschitz maps D. Dilatation of sphere-valued mappings E4-Degrees of short maps between compact and noncompact manifolds 3 Metric Structures on Families of Metric Spaces A. Lipschitz and Hausdorff distance B. The noncompact case C. The Hausdorff-Lipschitz metric, quasi-isometries, and word metrics D-(.First-order metric invariants and ultralimits E_^ Convergence with control 3^ Convergence and Concentration of Metrics and MeasuresA. A review of measures and mm spaces B.

Flow by mean curvature of convex surfaces into spheres

1984-01-01

Gerhard Huisken

The motion of surfaces by their mean curvature has been studied by Brakke [1] from the viewpoint of geometric measure theory. Other authors investigated the corresponding nonparametric problem [2], [5], … The motion of surfaces by their mean curvature has been studied by Brakke [1] from the viewpoint of geometric measure theory. Other authors investigated the corresponding nonparametric problem [2], [5], [9]. A reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal. In this paper we take a more classical point of view: Consider a compact, uniformly convex w-dimensional surface M = Mo without boundary, which is smoothly imbedded in R. Let Mo be represented locally by a diffeomorphism

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The heat equation shrinks embedded plane curves to round points

1987-01-01

M. Grayson

Soit C(•,0):S 1 →R 2 une courbe lisse plongee dans le plan. Alors C:S 1 ×[0,T)→R 2 existe en satisfaisant δC/δt=K•N, ou K est la courbure de C, et N … Soit C(•,0):S 1 →R 2 une courbe lisse plongee dans le plan. Alors C:S 1 ×[0,T)→R 2 existe en satisfaisant δC/δt=K•N, ou K est la courbure de C, et N est son vecteur unite normal entrant. C(•,t) est lisse pour tout t, il converge vers un point quand t\T et sa forme limite quand t→T est un cercle rond, avec convergence dans norme C ∞

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Contact Manifolds in Riemannian Geometry

1976-01-01

David E. Blair

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Three-manifolds with positive Ricci curvature

1982-01-01

Richard S. Hamilton

The evolution equation 259 4. Solution for a short time 260 5. Evolution equations with an integrability condition 262 6. Weakly parabolic linear systems 265 7. Evolution of the curvature … The evolution equation 259 4. Solution for a short time 260 5. Evolution equations with an integrability condition 262 6. Weakly parabolic linear systems 265 7. Evolution of the curvature 273 8. Curvature in dimension three 276 9. Preserving positive Ricci curvature .27910.Pinching the eigenvalues 283 11.The gradient of the scalar curvature 286 12. Interpolation inequalities for tensors 291 13.Higher derivatives of the curvature .29414.Long time existence 296 15.Controlling R^/R^ 299 16.Estimating the normalized equation 300 17. Exponential convergence 301

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The heat equation shrinking convex plane curves

1986-01-01

Michael E. Gage , Richard S. Hamilton

Let M and M' be Riemannian manifolds and F: M -» M' a smooth map Let M and M' be Riemannian manifolds and F: M -» M' a smooth map

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The Self-Duality Equations on a Riemann Surface

1987-07-01

Nigel Hitchin

Four-manifolds with positive curvature operator

1986-01-01

Richard S. Hamilton

A compact surface with positive mean scalar curvature must be diffeomorphic to the sphere S 2 or the real projective space RP 2 .A compact three-manifold with positive Ricci curvature … A compact surface with positive mean scalar curvature must be diffeomorphic to the sphere S 2 or the real projective space RP 2 .A compact three-manifold with positive Ricci curvature must be diffeomorphic to the sphere S 3 or a quotient of it by a finite group of fixed point free isometries in the standard metric, such as the real projective space RP 3 or a lens space L 3 p q .This was proven in [1].Our main result is the following generalization to four dimensions. Theorem.A compact four-manifold with a positive curvature operator is diffeomorphic to the sphere S 4 or the real projective space RP 4 .Here we regard the Riemannian curvature tensor Rm = {R iJkl } as a symmetric bilinear form on the two-forms Λ 2 by lettingWe say the manifold has a positive curvature operator if Rm(φ, φ) > 0 for all two-forms φ Φ 0, and a nonnegative curvature operator if Rm(φ,φ) ^ 0 for all φ.These results extend to the case of nonnegative curvature.A compact surface with nonnegative mean scalar curvature must be diffeomorphic to a quotient of the sphere S 2 or the plane R 2 by a group of fixed-point free isometries in the standard metrics.The examples are the sphere S 2 , the real projective space RP 2 , the torus T 2 = S ι X S\ and the Klein bottle(where # denotes the connected sum). Theorem.A compact three-manifold with nonnegative Ricci curvature is diffeomorphic to a quotient of one of the spaces S 3 or S 2 X R ι or R 3 by a group of fixed point free isometries in the standard metrics.The quotients of S 2 X R 1 include S 2 X S\ RP 2 X S\ the unoriented S 2 bundle over S\ and the connected sum K 3 = RP 3 #RP 3 .The quotients of R 3 are the torus T 3 and five other flat three-manifolds.

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Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

1999-02-19

Alexander Grigor’yan

We provide an overview of such properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and non-explosion. It is shown that both properties have various analytic characterizations, … We provide an overview of such properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and non-explosion. It is shown that both properties have various analytic characterizations, in terms of the heat kernel, Green function, Liouville properties, etc. On the other hand, we consider a number of geometric conditions such as the volume growth, isoperimetric inequalities, curvature bounds, etc., which are related to recurrence and non-explosion.

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On the structure of spaces with Ricci curvature bounded below. I

1997-01-01

Jeff Cheeger , Tobias Colding

Fix p and define the renormalized volume function, Fix p and define the renormalized volume function,

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Harmonic functions on complete riemannian manifolds

1975-03-01

Shing‐Tung Yau

Barycenters in the Wasserstein Space

2011-01-01

Martial Agueh , Guillaume Carlier

In this paper, we introduce a notion of barycenter in the Wasserstein space which generalizes McCann's interpolation to the case of more than two measures. We provide existence, uniqueness, characterizations, … In this paper, we introduce a notion of barycenter in the Wasserstein space which generalizes McCann's interpolation to the case of more than two measures. We provide existence, uniqueness, characterizations, and regularity of the barycenter and relate it to the multimarginal optimal transport problem considered by Gangbo and Święch in [Comm. Pure Appl. Math., 51 (1998), pp. 23–45]. We also consider some examples and, in particular, rigorously solve the Gaussian case. We finally discuss convexity of functionals in the Wasserstein space.

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Ricci curvature for metric-measure spaces via optimal transport

2009-04-22

John Lott , Cédric Villani

We define a notion of a measured length space X having nonnegative N -Ricci curvature, for N ∈ [1, ∞), or having ∞-Ricci curvature bounded below by K, for K … We define a notion of a measured length space X having nonnegative N -Ricci curvature, for N ∈ [1, ∞), or having ∞-Ricci curvature bounded below by K, for K ∈ R. The definitions are in terms of the displacement convexity of certain functions on the associated Wasserstein metric space P 2 (X) of probability measures.We show that these properties are preserved under measured Gromov-Hausdorff limits.We give geometric and analytic consequences.This paper has dual goals.One goal is to extend results about optimal transport from the setting of smooth Riemannian manifolds to the setting of length spaces.A second goal is to use optimal transport to give a notion for a measured length space to have Ricci curvature bounded below.We refer to [11] and [44] for background material on length spaces and optimal transport, respectively.Further bibliographic notes on optimal transport are in Appendix F. In the present introduction we motivate the questions that we address and we state the main results.To start on the geometric side, there are various reasons to try to extend notions of curvature from smooth Riemannian manifolds to more general spaces.A fairly general setting is that of length spaces, meaning metric spaces (X, d) in which the distance between two points equals the infimum of the lengths of curves joining the points.In the rest of this introduction we assume that X is a compact length space.Alexandrov gave a good notion of a length space having "curvature bounded below by K", with K a real number, in terms of the geodesic triangles in X.In the case of a Riemannian manifold M with the induced length structure, one recovers the Riemannian notion of having sectional curvature bounded below by K. Length spaces with Alexandrov curvature bounded below by K behave nicely with respect to the Gromov-Hausdorff topology on compact metric spaces (modulo isometries); they form a closed subset.

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On the parabolic kernel of the Schrödinger operator

1986-01-01

Peter Li , Shing Tung Yau

In this paper, we will study parabolic equations of the typeon a general Riemannian manifold.The function q(x, t) is assumed to be C 2 in the first variable and C … In this paper, we will study parabolic equations of the typeon a general Riemannian manifold.The function q(x, t) is assumed to be C 2 in the first variable and C 1 in the second variable.In classical situations [20], a Harnack inequality for positive solutions was established locally.However, the geometric dependency of the estimates is complicated and sometimes unclear.Our goal is to prove a Harnack inequality for positive solutions of (0.1) (w 2) by utilizing a gradient estimate derived in w 1.The method of proof is originated in [26] and [8], where they have studied the elliptic case, i.e. the solution is time independent.In some situations (Theorems 2.2 and

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The entropy formula for the Ricci flow and its geometric applications

2002-01-01

Grisha Perelman

We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric … We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric applications are given. In particular, (1) Ricci flow, considered on the space of riemannian metrics modulo diffeomorphism and scaling, has no nontrivial periodic orbits (that is, other than fixed points); (2) In a region, where singularity is forming in finite time, the injectivity radius is controlled by the curvature; (3) Ricci flow can not quickly turn an almost euclidean region into a very curved one, no matter what happens far away. We also verify several assertions related to Richard Hamilton's program for the proof of Thurston geometrization conjecture for closed three-manifolds, and give a sketch of an eclectic proof of this conjecture, making use of earlier results on collapsing with local lower curvature bound.

Computing Discrete Minimal Surfaces and Their Conjugates

1993-01-01

Ulrich Pinkall , Konrad Polthier

Abstract We present a new algorithm to compute stable discrete minimal surfaces bounded by a number of fixed or free boundary curves in R 3, S 3 and H 3. … Abstract We present a new algorithm to compute stable discrete minimal surfaces bounded by a number of fixed or free boundary curves in R 3, S 3 and H 3. The algorithm makes no restr iction on the genus and can handl e singular triangulations. Additionally, we present an algorithm that, starting from a discrete harmonic map, gives a conjugate harmonic map. This can be applied to the identity map on a minimal surface to produce its conjugate minimal surface, a procedure that often yields unstable solutions to a free boundary value problem for minimal surfaces. Symmetry properties of boundary curves are respected during conjugation.

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Real hypersurfaces in complex manifolds

1974-01-01

Shiing-Shen Chern , Jürgen Moser

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On the variational principle

1974-08-01

Ivar Ekeland

Asymptotic behavior for singularities of the mean curvature flow

1990-01-01

Gerhard Huisken

Let M n9 n > 1, be a compact «-dimensional manifold without boundary and assume that Fo: M n -> U n+{ smoothly immerses M n as a hypersurface in … Let M n9 n > 1, be a compact «-dimensional manifold without boundary and assume that Fo: M n -> U n+{ smoothly immerses M n as a hypersurface in a Euclidean (n + l)-space R π+1 .We say that MQ = F 0 (Af") is moved along its mean curvature vector if there is a whole family F( , t) of smooth immersions with corresponding hypersurfaces M t = F( , t)(M n ) such that

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Newtonian spaces: An extension of Sobolev spaces to metric measure spaces

2000-08-31

Nageswari Shanmugalingam

This paper studies a possible definition of Sobolev spaces in abstract metric spaces and answers in the affirmative the question whether this definition yields a Banach space. The paper also … This paper studies a possible definition of Sobolev spaces in abstract metric spaces and answers in the affirmative the question whether this definition yields a Banach space. The paper also explores the relationship between this definition and the Hajlasz spaces. For specialized metric spaces the Sobolev embedding theorems are proven. Different versions of capacities are also explored and these various definitions are compared. The main tool used in this paper is the concept of moduli of path families.

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Generalized gradients and applications

1975-01-01

Frank H. Clarke

A theory of generalized gradients for a general class of functions is developed, as well as a corresponding theory of normals to arbitrary closed sets.It is shown how these concepts … A theory of generalized gradients for a general class of functions is developed, as well as a corresponding theory of normals to arbitrary closed sets.It is shown how these concepts subsume the usual gradients and normals of smooth functions and manifolds, and the subdifferentials and normals of convex analysis.A theorem is proved concerning the differentiability properties of a function of the form max{g(x, u):u e if}.This result unifies and extends some theorems of Danskin and others.The results are then applied to obtain a characterization of flow-invariant sets which yields theorems of Bony and Brezis as corollaries.

Curvatures of left invariant metrics on lie groups

1976-09-01

John Milnor

On the geometry of metric measure spaces. II

2006-01-01

Karl‐Theodor Sturm

We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$ (introduced in [Sturm K-T … We introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound $\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m} \right)} > K$ (introduced in [Sturm K-T (2006) On the geometry of metric measure spaces. I. Acta Math 196:65–131]) which is recovered as the borderline case CD(K, ∞). The additional real parameter N plays the role of a generalized upper bound for the dimension. For Riemannian manifolds, CD(K, N) is equivalent to ${\text{Ric}}_{M} {\left( {\xi ,\xi } \right)} > K{\left| \xi \right|}^{2} $ and dim(M) ⩽ N. The curvature-dimension condition CD(K, N) is stable under convergence. For any triple of real numbers K, N, L the family of normalized metric measure spaces (M, d, m) with CD(K, N) and diameter ⩽ L is compact. Condition CD(K, N) implies sharp version of the Brunn–Minkowski inequality, of the Bishop–Gromov volume comparison theorem and of the Bonnet–Myers theorem. Moreover, it implies the doubling property and local, scale-invariant Poincaré inequalities on balls. In particular, it allows to construct canonical Dirichlet forms with Gaussian upper and lower bounds for the corresponding heat kernels.

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On Isolated Rational Singularities of Surfaces

1966-01-01

Michael Artin

In 1934, DuVal [3] listed the configurations of curves which can be obtained by resolving certain isolated double points of embedded surfaces (they are depicted in the figure below). These … In 1934, DuVal [3] listed the configurations of curves which can be obtained by resolving certain isolated double points of embedded surfaces (they are depicted in the figure below). These configurations arise naturally in other contexts, for instance as exceptional curves for pluricanonical embeddings of surfaces [7], and so it seems desirable to have a converse result, showing that a singularity giving rise to such a configuration is necessarily a double point. We have reconsidered the question in a more general context, and obtain in addition the correct nunferical characterization of singularities (cf. definition below). This characterization is made without the assumption that the surface is embedded and points out the connection of Du Val's work with Castelnuovo's criterion for exceptional curves [2], ([8], p. 38). Finally, we list the configurations obtained from rational triple points.

On the proof of the positive mass conjecture in general relativity

1979-02-01

Richard Schoen , Shing-Tung Yau

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11. The Imbedding Problem for Riemannian Manifolds

2002-12-31

JohnHG Nash

THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION

2001-01-31

Félix Otto

Abstract We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically … Abstract We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show that the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior. Keywords: AMS code: 35K55; 76S05; 58B20; 58F39 ACKNOWLEDGMENTS The author's PhD advisor, Stephan Luckhaus, always stressed the importance of underlying gradient flow structures. David Kinderlehrer is coauthor on an earlier paper on the relationship between diffusion equations and the Wasserstein metric, this paper owes to many conversations with him. The idea of using Riemannian calculus to derive contraction properties for partial differential equations developed in conversations with Robert McCann in 1997 and later with Cedric Villani in 1998. The author gave a lecture series on the results of this paper in 1998 at the Max–Planck Institute for Mathematics in the Sciences in Leipzig, and wishes to thank the Institute for this opportunity.

Manifolds of negative curvature

1969-01-01

Richard L. Bishop , Barrett O’Neill

IntroductionACm function/on a riemannian manifold M is convex provided its hessian (second covariant differential) is positive semidefinite, or equivalently if (/o <t)"5:0 for every geodesic a in M. We shall … IntroductionACm function/on a riemannian manifold M is convex provided its hessian (second covariant differential) is positive semidefinite, or equivalently if (/o <t)"5:0 for every geodesic a in M. We shall apply this notion in a variety of ways to the study of manifolds of negative or nonpositive curvature.Convexity has, of course, long been associated with negative curvature, but convex functions seem to have been used only locally or along curves.In the first part of this paper we give an abstract global treatment.Nonconstant convex functions exist only on manifolds of infinite volume (2.2); the first question about such a function on M (complete, A'á 0) is whether it has a critical point-necessarily an absolute minimum.If not, M is diffeomorphic to a product LxR1 (3.12).If so, much of the topology and geometry of M is determined by the minimum set C of/.This comes about as follows.Like any set {me M \ f(m)f¿a}, C is totally convex, that is, contains a geodesic segment a whenever it contains the endpoints of a.Let A he an arbitrary closed, totally convex set in M. In case A is a submanifold, it is totally geodesic and M is, via exponentiation, its normal bundle (3.1).This situation does not change greatly if A is not a submanifold (e.g., 3.4); A is always a topological manifold with boundary (possibly nonsmooth), whose interior is a locally totally geodesic submanifold.We describe a number of geometrically significant ways of constructing convex functions (4.1, 4.2, 4.8, 5.5, etc.); these show in particular that C may or may not be a submanifold.In the second part of the paper we define and study the mobility sequence of a nonpositive curvature manifold M. The basic fact is that the set P(M) of common zeroes of all Killing fields on M is a closed, totally convex submanifold (5.1).Thus M is a vector bundle over P(M), which is totally geodesic and hence again has K^O.The mobility sequence is then constructed by iteration: M^¡*P(M)■ ■ ■ -¡-Pk(M)=Q.It terminates with a submanifold that is either mobile (P(Q) empty) or immobile (P(Q) = Q).We prove that if Q is mobile, or if ttx(M) has nontrivial center, then (with a trivial exception) M is a product LxR1 and if also M contains a closed geodesic then in particular M is a vector bundle over a circle (4.9, 6.4).Since P(M) is invariant under all isometries of M, the mobility sequence is closely related to the isometry group of M (e.g.8.1).We introduce the notion of warped product (or, more generally, warped bundle),

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Positive scalar curvature and the Dirac operator on complete riemannian manifolds

1983-12-01

Mikhael Gromov , H. Blaine Lawson

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R-Torsion and the Laplacian on Riemannian manifolds

1971-10-01

Daniel Ray , I. M. Singer

Motion of level sets by mean curvature. I

1991-01-01

L. C. Evans , Joel Spruck

We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature.This weak solution allows us then to define … We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature.This weak solution allows us then to define for any compact set Γ o a unique generalized motion by mean curvature, existing for all time.We investigate the various geometric properties and pathologies of this evolution.

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On the First Variation of a Varifold

1972-05-01

William K. Allard

Suppose M is a smooth m dimensional Riemannian manifold and k is a positive integer not exceeding m. Our purpose is to study the first variation of the k dimensional … Suppose M is a smooth m dimensional Riemannian manifold and k is a positive integer not exceeding m. Our purpose is to study the first variation of the k dimensional area integrand in M. Our main result is a regularity theorem for weakly defined k dimensional surfaces in M whose first variation of area is summable to a power greater than k. A natural domain for any k dimensional parametric integral in M, among which the simplest is the k dimensional area integral, is the space of k dimensional varifolds in M introduced by Almgren in [AF 1]. Such a varifold is defined to be a Radon measure on the bundle over M whose fiber at each point p of M is the Grassmann manifold of k dimensional linear subspaces of the tangent space to M at p. If V is a varifold in M, let I I V I I be the Radon measure on M obtained from V by ignoring the fiber variables. Naturally injected in the space of k dimensional varifolds in M is the set of k dimensional rectifiable subsets of M, which includes the set of k dimensional submanifolds of M as well as more general k dimensional surfaces in M. A k dimensional varifold in M is said to be rectifiable (integral) if it can be strongly approximated by a positive real (integral) linear combination of varifolds corresponding to continuously differentiable k dimensional submanifolds of M. To any classical k dimensional geometric object in M there corresponds a k dimensional integral varifold in M. If N is a smooth Riemannian manifold and F: M-e N is smooth, then F induces in a natural way a strongly continuous mapping F# of the k dimen-

The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality

2001-11-01

Gerhard Huisken , Tom Ilmanen

Let M be an asymptotically flat 3-manifold of nonnegative scalar curvature. The Riemannian Penrose Inequality states that the area of an outermost minimal surface N in M is bounded by … Let M be an asymptotically flat 3-manifold of nonnegative scalar curvature. The Riemannian Penrose Inequality states that the area of an outermost minimal surface N in M is bounded by the ADM mass m according to the formula |N| ≤ 16πm2. We develop a theory of weak solutions of the inverse mean curvature flow, and employ it to prove this inequality for each connected component of N using Geroch's monotonicity formula for the ADM mass. Our method also proves positivity of Bartnik's gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry.

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Spectral asymmetry and Riemannian Geometry. I

1975-01-01

Michael Atiyah , V. K. Patodi , I. M. Singer

1. Introduction . The main purpose of this paper is to present a generalization of Hirzebruch's signature theorem for the case of manifolds with boundary. Our result is in the … 1. Introduction . The main purpose of this paper is to present a generalization of Hirzebruch's signature theorem for the case of manifolds with boundary. Our result is in the framework of Riemannian geometry and can be viewed as analogous to the Gauss–Bonnet theorem for manifolds with boundary, although there is a very significant difference between the two cases which is, in a sense, the central topic of the paper. To explain this difference let us begin by recalling that the classical Gauss–Bonnet theorem for a surface X with boundary Y asserts that the Euler characteristic E(X) is given by a formula: where K is the Gauss curvature of X and σ is the geodesic curvature of Y in X . In particular if, near the boundary, X is isometric to the product Y x R + , the boundary integral in (1.1) vanishes and the formula is the same as for closed surfaces. Similar remarks hold in higher dimensions. Now if X is a closed oriented Riemannian manifold of dimension 4, there is another formula relating cohomological invariants with curvature in addition to the Gauss–Bonnet formula. This expresses the signature of the quadratic form on H 2 ( X , R) by an integral formula where p 1 is the differential 4-form representing the first Pontrjagin class and is given in terms of the curvature matrix R by p 1 = (2π) −2 Tr R 2 . It is natural to ask if (1.2) continues to hold for manifolds with boundary, provided the metric is a product near the boundary. Simple examples show that this is false, so that in general

Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

2000-06-01

Félix Otto , Cédric Villani

Some mathematical questions related to the mhd equations

1983-09-01

Michel Sermange , Roger Témam

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Conformal deformation of a Riemannian metric to constant scalar curvature

1984-01-01

Richard Schoen

A well-known open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This problem is … A well-known open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This problem is known as the Yamabe problem because it was formulated by Yamabe [8] in 1960, While Yamabe's paper claimed to solve the problem in the affirmative, it was found by N. Trudinger [6] in 1968 that Yamabe's paper was seriously incorrect. Trudinger was able to correct Yamabe's proof in case the scalar curvature is nonpositive. Progress was made on the case of positive scalar curvature by T. Aubin [1] in 1976. Aubin showed that if dim M > 6 and M is not conformally flat, then M can be conformally changed to constant scalar curvature. Up until this time, Aubin's method has given no information on the Yamabe problem in dimensions 3, 4, and 5. Moreover, his method exploits only the local geometry of M in a small neighborhood of a point, and hence could not be used on a conformally flat manifold where the Yamabe problem is clearly a global problem. Recently, a number of geometers have been interested in the conformally flat manifolds of positive scalar curvature where a solution of Yamabe's problem gives a conformally flat metric of constant scalar curvature, a metric of some geometric interest. Note that the class of conformally flat manifolds of positive scalar curvature is closed under the operation of connected sum, and hence contains connected sums of spherical space forms with copies of S X S~. In this paper we introduce a new global idea into the problem and we solve it in the affirmative in all remaining cases; that is, we assert the existence of a positive solution u on M of the equation

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Ricci flow with surgery on three-manifolds

2003-01-01

Grisha Perelman

This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that … This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that e-print; the exceptions are (1) the statement that manifolds that can collapse with local lower bound on sectional curvature are graph manifolds - this is deferred to a separate paper, since the proof has nothing to do with the Ricci flow, and (2) the claim on the lower bound for the volume of maximal horns and the smoothness of solutions from some time on, which turned out to be unjustified and, on the other hand, irrelevant for the other conclusions.

Asymptotics for a Class of Non-Linear Evolution Equations, with Applications to Geometric Problems

1983-11-01

Leon Simon

Soit Σ une variete de Riemann compacte et soit une fonction reguliere u=u(x,t), (x,t)∈ΣX(0,T) (T>0) satisfaisant une equation d'evolution soit de la forme ci-#7B-M(u)=f soit de la forme −u+u˙-#7B-M(u)-#7B-R(u)=f. (#7B-M(u) … Soit Σ une variete de Riemann compacte et soit une fonction reguliere u=u(x,t), (x,t)∈ΣX(0,T) (T>0) satisfaisant une equation d'evolution soit de la forme ci-#7B-M(u)=f soit de la forme −u+u˙-#7B-M(u)-#7B-R(u)=f. (#7B-M(u) est l'operateur d'Euler-Lagrange d'ordre 2). On etudie le comportement asymptotique des solutions de ces equations

Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations

1991-01-01

Yun Gang Chen , Yoshikazu Giga , Shun’ichi Goto

This paper treats degenerate parabolic equations of second order $$u_t + F(\nabla u,\nabla ^2 u) = 0$$ (14.1) related to differential geometry, where ∇ stands for spatial derivatives of u … This paper treats degenerate parabolic equations of second order $$u_t + F(\nabla u,\nabla ^2 u) = 0$$ (14.1) related to differential geometry, where ∇ stands for spatial derivatives of u = u{t,x) in x ∈ R n , and u t represents the partial derivative of u in time t. We are especially interested in the case when (1.1) is regarded as an evolution equation for level surfaces of u. It turns out that (1.1) has such a property if F has a scaling invariance $$F(\lambda p,\lambda X + \sigma p \otimes p) = \lambda F(p.X),\,\,\,\,\,\,\lambda > 0,\,\,\sigma \in \mathbb{R}$$ (14.2) for a nonzero p ∈ R n and a real symmetric matrix X, where ⊗ denotes a tensor product of vectors in R n . We say (1.1) is geometric if F satisfies (1.2). A typical example is $$u_t - \left| {\nabla u} \right|div(\nabla u/\left| {\nabla u} \right|) = 0,$$ (14.3) where ∇u is the (spatial) gradiant of u. Here ∇u/|∇u| is a unit normal to a level surface of u, so div (∇u/|∇u|) is its mean curvature unless ∇u vanishes on the surface. Since u t /\∇u is a normal velocity of the level surface, (1.3) implies that a level surface of solution u of (1.3) moves by its mean curvature unless ∇u vanishes on the surface. We thus call (1.3) the mean curvature flow equation in this paper.

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Remarks concerning the conformal deformation of riemannian structures on compact manifolds

1968-01-01

Neil S. Trudinger

Riemannian Geometry of Contact and Symplectic Manifolds

2002-01-01

David E. Blair

The author's lectures, "Contact Manifolds in Riemannian Geometry," volume 509 (1976), in the Springer-Verlag Lecture Notes in Mathematics series have been out of print for some time and it seems … The author's lectures, "Contact Manifolds in Riemannian Geometry," volume 509 (1976), in the Springer-Verlag Lecture Notes in Mathematics series have been out of print for some time and it seems appro

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Singular hypersurfaces and thin shells in general relativity

1967-04-01

W. Israel

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Invariant Manifolds

1977-01-01

Morris W. Hirsch , Charles Pugh , Michael Shub

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On the geometry of metric measure spaces

2006-01-01

Karl‐Theodor Sturm

We introduce and analyze lower (Ricci) curvature bounds $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾ K for metric measure spaces $ {\left( {M,d,m} \right)} $. Our definition is based on … We introduce and analyze lower (Ricci) curvature bounds $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾ K for metric measure spaces $ {\left( {M,d,m} \right)} $. Our definition is based on convexity properties of the relative entropy $ Ent{\left( { \cdot \left| m \right.} \right)} $ regarded as a function on the L2-Wasserstein space of probability measures on the metric space $ {\left( {M,d} \right)} $. Among others, we show that $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds, $ \underline{{Curv}} {\left( {M,d,m} \right)} $ ⩾ K if and only if $ Ric_{M} {\left( {\xi ,\xi } \right)} $ ⩾ K$ {\left| \xi \right|}^{2} $ for all $ \xi \in TM $. The crucial point is that our lower curvature bounds are stable under an appropriate notion of D-convergence of metric measure spaces. We define a complete and separable length metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation. We also prove that the family of normalized metric measure spaces with doubling constant ⩽ C is closed under D-convergence. Moreover, the family of normalized metric measure spaces with doubling constant ⩽ C and diameter ⩽ L is compact under D-convergence.

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The Ricci flow on surfaces

1988-01-01

Richard S. Hamilton

Neck pinch singularities and Joyce conjectures in Lagrangian mean curvature flow with circle symmetry

2025-06-26

Jason D. Lotay , Gonçalo Oliveira | Journal of the European Mathematical Society

In this article, we consider Lagrangian mean curvature flow of compact, circle-invariant, almost calibrated Lagrangian surfaces in hyperkähler 4-manifolds with circle symmetry. We show that this Lagrangian mean curvature flow … In this article, we consider Lagrangian mean curvature flow of compact, circle-invariant, almost calibrated Lagrangian surfaces in hyperkähler 4-manifolds with circle symmetry. We show that this Lagrangian mean curvature flow can be continued for all time, through a finite number of finite time singularities, and converges to a chain of special Lagrangians, verifying various aspects of Joyce’s conjectures, Joyce (2015), in this setting. This result provides the first non-trivial setting where Lagrangian mean curvature flow may be used successfully to achieve the desired decomposition of a Lagrangian into a sum of special Lagrangians representing its Hamiltonian isotopy class. We also show that the singularities of the flow are neck pinches in the sense conjectured by Joyce and give examples where such finite time singularities are guaranteed to occur.

Results on the elliptic Sombor index

2025-06-24

Fateme Movahedi | RAIRO - Operations Research

‎The newly introduced elliptic Sombor index of a graph $G$‎‏ ‎is defined as follows‎‎[[EQUATION]]‎ ‎in which $E$‎‏ ‎and $d(‏‎u‎)$ are the edge set of $G$‎‏ ‎and the degree of the … ‎The newly introduced elliptic Sombor index of a graph $G$‎‏ ‎is defined as follows‎‎[[EQUATION]]‎ ‎in which $E$‎‏ ‎and $d(‏‎u‎)$ are the edge set of $G$‎‏ ‎and the degree of the vertex $u$ in $G$‎‏, ‎respectively‎. ‎\\‎ ‎In this paper‎, ‎we compute the elliptic Sombor index for certain graphs. Furthermor‏e, we obtain new results and bounds ‎for‎ the elliptic Sombor index in a graph‎.

Some remarks on the moment curves in R4 and R3

2025-06-24

Mircea Crasmareanu | Istanbul Journal of Mathematics

On the structure at infinity of complete smooth metric measure spaces with a weighted Poincaré inequality

2025-06-24

Ha Tuan Dung | manuscripta mathematica

A Characterization of Horizontally Totally Geodesic Hypersurfaces in Heisenberg Groups

2025-06-24

Andrea Pinamonti , Simone Verzellesi | Journal of Geometric Analysis

Width Stability of Rotationally Symmetric Metrics

2025-06-24

Hunter Stufflebeam , Paul Sweeney | Journal of Geometric Analysis

Abstract In 2018, Marques and Neves proposed a volume preserving intrinsic flat stability conjecture concerning their width rigidity theorem for the unit round 3-sphere. In this work, we establish the … Abstract In 2018, Marques and Neves proposed a volume preserving intrinsic flat stability conjecture concerning their width rigidity theorem for the unit round 3-sphere. In this work, we establish the validity of this conjecture under the additional assumption of rotational symmetry. Furthermore, we obtain a rigidity theorem in dimensions at least three for rotationally symmetric manifolds, which is analogous to the width rigidity theorem of Marques and Neves. We also prove a volume preserving intrinsic flat stability result for this rigidity theorem. Lastly, we study variants of Marques and Neves’ stability conjecture. In the first, we show Gromov–Hausdorff convergence outside of certain “bad” sets. In the second, we assume non-negative Ricci curvature and show Gromov–Hausdorff stability.

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None

2025-06-23

Matthew Rosenzweig , Sylvia Serfaty | Annales de la faculté des sciences de Toulouse Mathématiques

We consider mean-field limits for overdamped Langevin dynamics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> particles with possibly singular interactions. It has been shown that a modulated free energy method can be used to … We consider mean-field limits for overdamped Langevin dynamics of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> particles with possibly singular interactions. It has been shown that a modulated free energy method can be used to prove the mean-field convergence or propagation of chaos for a certain class of interactions, including Riesz kernels. We show here that generation of chaos, i.e. exponential in time convergence to a tensorized (or iid) state starting from a nontensorized one, can be deduced from the modulated free energy method provided a uniform-in-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> “modulated logarithmic Sobolev inequality” holds. Proving such an inequality is a question of independent interest, which is generally difficult. As an illustration, we show that uniform modulated logarithmic Sobolev inequalities can be proven for a class of situations in one dimension.

Very weak solutions of the Dirichlet problem for 2-Hessian equation

2025-06-23

Tongtong Li , Guohuan Qiu | Journal of Differential Equations

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Small eigenvalues of the Hodge-Laplacian with sectional curvature bounded below

2025-06-23

Colette Anné , Junya Takahashi | Annals of Global Analysis and Geometry

Certain contact metric as $$\eta$$-Ricci soliton

2025-06-23

Amalendu Ghosh | The Journal of Analysis

Chen inequalities for immersions in trans-Sasakian space forms with slant factor

2025-06-22

Mohd Danish Siddiqi , Aliya Naaz Sıddıquı | Carpathian Mathematical Publications

In this research article, our focus is directed towards the exploration of trans-Sasakian manifolds that incorporate a distinctive type of non-metric connection referred to as a quarter-symmetric non-metric ($QSNM$) connection. … In this research article, our focus is directed towards the exploration of trans-Sasakian manifolds that incorporate a distinctive type of non-metric connection referred to as a quarter-symmetric non-metric ($QSNM$) connection. We delve into the derivation of the mathematical expressions governing the curvature tensor $\widetilde{R}$ of trans-Sasakian space forms, utilizing the aforementioned $QSNM$-connection. Our primary efforts are centered around the establishment of Chen inequalities. These inequalities find application in the characterization of slant submanifolds in the trans-Sasakian space forms and connected by a $QSNM$-connection. Furthermore, our investigation encompasses the classification of Chen invariants. This classification is extended to $\alpha$-Sasakian, $\beta$-Kenmotsu and cosymplectic manifolds, all of which are endowed with the distinctive $QSNM$-connection.

Weak Nearly Kenmotsu Manifolds and Gradient Ricci Solitons

2025-06-22

Sourav Nayak , Dhriti Sundar Patra , Vladimir Rovenski | Mediterranean Journal of Mathematics

On Hardy-Poincaré Inequalities

2025-06-21

Tohru Ozawa , Durvudkhan Suragan | Journal of Mathematical Sciences

Notes on the uniqueness of Type II Yamabe metrics

2025-06-20

Shota Hamanaka , Pak Tung Ho | Nonlinear Differential Equations and Applications NoDEA

Characterizing bi-harmonic homomorphisms in three-dimensional unimodular Lie groups

2025-06-20

Kaddour Zegga , Abdelkader Zagane , Wassim Merchela | Vestnik Udmurtskogo Universiteta Matematika Mekhanika Komp yuternye Nauki

This study is dedicated to the classification of bi-harmonic homomorphisms $\varphi\colon(G,g)\to (H,h)$, where $G$ and $H$ represent connected and simply connected three-dimensional unimodular Lie groups, while $g$ and $h$ denote … This study is dedicated to the classification of bi-harmonic homomorphisms $\varphi\colon(G,g)\to (H,h)$, where $G$ and $H$ represent connected and simply connected three-dimensional unimodular Lie groups, while $g$ and $h$ denote left invariant Riemannian metrics.

Classification of Möbius homogeneous superconformal surfaces in 𝕊5

2025-06-20

Changping Wang , Zhenxiao Xie | Pacific Journal of Mathematics

On a global integral equation arising in the discretization of granular flows

2025-06-20

Adrian Tudorascu | Nonlinear Analysis

Fourier dimension of constant rank hypersurfaces

2025-06-19

Jing-yu Zhu | Mathematische Annalen

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A note on pointwise quasi hemi-slant submersions

2025-06-19

Sushil Kumar , Rajendra Prasad , Abdul Haseeb +1 more | Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics

As a generalization of hemi-slant and semi-slant submersions, we discuss pointwise quasi Hemi-slant (PQHS) submersions from almost Hermitian manifolds onto Riemannian manifolds. We obtain various results satisfied by these submersions … As a generalization of hemi-slant and semi-slant submersions, we discuss pointwise quasi Hemi-slant (PQHS) submersions from almost Hermitian manifolds onto Riemannian manifolds. We obtain various results satisfied by these submersions from Kähler manifolds onto Riemannian manifolds. Moreover, we find necessary and sufficient conditions on integrability of the distributions, and explore the geometry of totally geodesic foliations of discussed submersions. At last, we construct some examples of a PQHS submersion from an almost Hermitian manifold onto a Riemannian manifold.

CMC hypersurface with finite index in hyperbolic space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">H</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:math>

2025-06-18

Han Hong | Advances in Mathematics

Bi-Coupling method and applications

2025-06-18

Panpan Ren , Feng‐Yu Wang | Probability Theory and Related Fields

Double exceptional points in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mstyle mathvariant="script"><mml:mi>𝒫</mml:mi><mml:mi>𝒯</mml:mi></mml:mstyle></mml:math> -symmetric structures with an anisotropic layer

2025-06-18

NULL AUTHOR_ID | Physical review. B./Physical review. B

Constant Angle Surfaces in $$\mathbb {M}^2_1(\kappa )\times {\mathbb {R}}$$

2025-06-17

L Pellegrino | Mediterranean Journal of Mathematics

Classification of primitive immmersions of constant curvature into flag manifolds

2025-06-17

Rui Pacheco , Mujeeb ur Rehman | Collectanea mathematica

Killing Reeb vector fields on some almost contact manifolds

2025-06-17

Fortuné Massamba | Mathematica Bohemica

Pseudo-Quasi Conformal Curvature Tensor of Quasi Sasakian Manifold with Generalized Sasakian Space Forms

2025-06-17

Farah Hasan AlHusseini , Habeeb M. Abood | Malaysian Journal of Mathematical Sciences

The current study focuses on three main topics; the pseudo-quasi conformal curvature tensor, quasi-Sasakian manifolds (QSAS-manifold), and generalized Sasakian space forms (GS-space forms) employing the G-adjoined structure space. For a … The current study focuses on three main topics; the pseudo-quasi conformal curvature tensor, quasi-Sasakian manifolds (QSAS-manifold), and generalized Sasakian space forms (GS-space forms) employing the G-adjoined structure space. For a QSAS-manifold of GS-space forms, the components of the Ricci tensor and pseudo-quasi conformal curvature tensor are computed. Various types of QSAS-manifold are described, and their interactions with GS-space forms are investigated. It has been shown that ξ-pseudo quasi conformally flat QSAS-manifold of GSspace forms includes quasi Einstein manifold. Furthermore, the condition of a quasi-pseudo quasi conformal QSAS-manifold of space forms to be a quasi-Einstein manifold is identified. Finally, the scalar curvature is determined for ξ-pseudo quasi-conformally flat QSAS-manifold and quasi-pseudo quasi conformal QSAS-manifold.

None

2025-06-17

Dimitri Navarro | Annales de l’institut Fourier

This paper focuses on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">RCD</mml:mi><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>-spaces, i.e. possibly non-smooth metric measure spaces with nonnegative Ricci curvature and dimension at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math>. First, we establish a list of the … This paper focuses on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">RCD</mml:mi><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>-spaces, i.e. possibly non-smooth metric measure spaces with nonnegative Ricci curvature and dimension at most <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn></mml:math>. First, we establish a list of the compact topological spaces admitting an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">RCD</mml:mi><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>-structure. We then describe the moduli space of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">RCD</mml:mi><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>-structures for each space and show that it is contractible.

Some Results on Radical Screen Transversal Lightlike Submanifold of an Indefinite Kaehler Norden Manifold

2025-06-16

Anu Devgan | International Journal for Research in Applied Science and Engineering Technology

In this paper, we introduce the notion of radical screen transversal and screen transversal anti-invariant lightlike submanifolds of an indefinite Kaehler Norden manifold. We investigate the geometry of distributions involved … In this paper, we introduce the notion of radical screen transversal and screen transversal anti-invariant lightlike submanifolds of an indefinite Kaehler Norden manifold. We investigate the geometry of distributions involved and obtain necessary and sufficient conditions for the induced connection on radical screen transversal and screen transversal antiinvariant lightlike submanifolds to be metric connection. Further, we provide the necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic on radical screen transversal lightlike submanifold of an indefinite Kaehler Norden manifold.

Volumetric Minkowski Inequality on Manifolds with Weighted Poincaré Inequality

2025-06-16

Wing Kong Chiu , Chiung-Jue Anna Sung | Journal of Geometric Analysis

Examples for scalar sphere stability

2025-06-14

Paul Sweeney | Calculus of Variations and Partial Differential Equations

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Geometric and Structural Properties of Indefinite Kenmotsu Manifolds Admitting Eta-Ricci–Bourguignon Solitons

2025-06-14

Mohd. Aquib , Oḡuzhan Bahadır , Laltluangkima Chawngthu +1 more | Mathematics

This paper undertakes a detailed study of η-Ricci–Bourguignon solitons on ϵ-Kenmotsu manifolds, with particular focus on three special types of Ricci tensors: Codazzi-type, cyclic parallel and cyclic η-recurrent tensors that … This paper undertakes a detailed study of η-Ricci–Bourguignon solitons on ϵ-Kenmotsu manifolds, with particular focus on three special types of Ricci tensors: Codazzi-type, cyclic parallel and cyclic η-recurrent tensors that support such solitonic structures. We derive key curvature conditions satisfying Ricci semi-symmetric (R·E=0), conharmonically Ricci semi-symmetric (C(ξ,βX)·E=0), ξ-projectively flat (P(βX,βY)ξ=0), projectively Ricci semi-symmetric (L·P=0) and W5-Ricci semi-symmetric (W(ξ,βY)·E=0), respectively, with the admittance of η-Ricci–Bourguignon solitons. This work further explores the role of torse-forming vector fields and provides a thorough characterization of ϕ-Ricci symmetric indefinite Kenmotsu manifolds admitting η-Ricci–Bourguignon solitons. Through in-depth analysis, we establish significant geometric constraints that govern the behavior of these manifolds. Finally, we construct explicit examples of indefinite Kenmotsu manifolds that satisfy the η-Ricci–Bourguignon solitons equation, thereby confirming their existence and highlighting their unique geometric properties. Moreover, these solitonic structures extend soliton theory to indefinite and physically meaningful settings, enhance the classification and structure of complex geometric manifolds by revealing how contact structures behave under advanced geometric flows and link the pure mathematical geometry to applied fields like general relativity. Furthermore, η-Ricci–Bourguignon solitons provide a unified framework that deepens our understanding of geometric evolution and structure-preserving transformations.

Matrix Li–Yau–Hamilton estimates for nonlinear heat equations under geometric flows

2025-06-14

Sha Yao , Hao-Yue Liu , Xin-An Ren | Pacific Journal of Mathematics

A flow method for curvature equations

2025-06-14

Shanwei Ding , Guanghan Li | Nonlinear Analysis

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Translating solitons of the mean curvature flow in warped products: nonexistence and rigidity

2025-06-14

Cícero P. Aquino , Henrique F. de Lima , G. S. Dias | Bollettino dell Unione Matematica Italiana

Surprises in the ordinary: O(N) invariant surface defect in the ϵ-expansion

2025-06-13

Oleksandr Diatlyk , Zimo Sun , Yifan Wang | Journal of High Energy Physics

A bstract We study an O ( N ) invariant surface defect in the Wilson-Fisher conformal field theory (CFT) in d = 4 – ϵ dimensions. This defect is defined … A bstract We study an O ( N ) invariant surface defect in the Wilson-Fisher conformal field theory (CFT) in d = 4 – ϵ dimensions. This defect is defined by mass deformation on a two-dimensional surface that generates localized disorder and is conjectured to factorize into a pair of ordinary boundary conditions in d = 3. We determine defect CFT data associated with the lightest O ( N ) singlet and vector operators up to the third order in the ϵ -expansion, find agreements with results from numerical methods and provide support for the factorization proposal in d = 3. Along the way, we observe surprising non-renormalization properties for surface anomalous dimensions and operator-product-expansion coefficients in the ϵ -expansion. We also analyze the full conformal anomalies for the surface defect.

Spectral estimates for free boundary minimal surfaces via Montiel–Ros partitioning methods

2025-06-13

Alessandro Carlotto , Mario B. Schulz , David Wiygul | Analysis & PDE

Asymptotic behaviour of unstable perturbations of the Fubini–Study metric in Ricci flow

2025-06-12

David Garfinkle , James Isenberg , Dan Knopf +1 more | Nonlinearity

Abstract Kröncke has shown that the Fubini–Study metric is an unstable generalised stationary solution of Ricci flow (Kröncke 2020 Commun. Anal. Geom. 28 35–394). In this paper, we carry out … Abstract Kröncke has shown that the Fubini–Study metric is an unstable generalised stationary solution of Ricci flow (Kröncke 2020 Commun. Anal. Geom. 28 35–394). In this paper, we carry out numerical simulations which indicate that Ricci flow solutions originating at unstable perturbations of the Fubini–Study metric develop local singularities modelled by the blowdown soliton discovered in (Feldman et al 2003 J. Differ. Geom. 65 169–209).

A property of homogeneous isoparametric submanifolds

2025-06-12

Cristián U. Sánchez | Revista de la Unión Matemática Argentina

On astheno-K ̈ahler manifolds

2025-06-11

Punam Gupta , Nidhi Yadav | Annals of the Alexandru Ioan Cuza University - Mathematics

This survey explores a range of classical findings and recent developments related to our understanding of astheno-K ̈ahler manifolds. Furthermore, we provide various examples of astheno-K ̈ahler manifolds and analyze … This survey explores a range of classical findings and recent developments related to our understanding of astheno-K ̈ahler manifolds. Furthermore, we provide various examples of astheno-K ̈ahler manifolds and analyze the challenges associated with their existence.

Real hypersurfaces in $$\mathbb {C}P^2$$ or $$\mathbb {C}H^2$$ whose shape operator commutes with the Cho operators

2025-06-10

Wenjie Wang | Periodica Mathematica Hungarica

Hessian estimates for Lagrangian mean curvature equation with sharp Lipschitz phase

2025-06-10

Xingchen Zhou | Calculus of Variations and Partial Differential Equations

Regularized unbalanced optimal transport as entropy minimization with respect to branching Brownian motion

2023-01-01

Aymeric Baradat , Hugo Lavenant

We consider the problem of minimizing the entropy of a law with respect to the law of a reference branching Brownian motion under density constraints at an initial and final … We consider the problem of minimizing the entropy of a law with respect to the law of a reference branching Brownian motion under density constraints at an initial and final time. We call this problem the branching Schr\odinger problem by analogy with the Schr\odinger problem, where the reference process is a Brownian motion. Whereas the Schr\odinger problem is related to regularized (a.k.a. entropic) optimal transport, we investigate here the link of the branching Schr\odinger problem with regularized unbalanced optimal transport. This link is shown at two levels. First, relying on duality arguments, the values of these two problems of calculus of variations are linked, in the sense that the value of the regularized unbalanced optimal transport (seen as a function of the initial and final measure) is the lower semi-continuous relaxation of the value of the branching Schr\odinger problem. Second, we also explicit a correspondence between the competitors of these two problems, and to that end we provide a fine description of laws having a finite entropy with respect to a reference branching Brownian motion. We investigate the small noise limit, when the noise intensity of the branching Brownian motion goes to $0$: in this case we show, at the level of the optimal transport model, that there is convergence to partial optimal transport. We also provide formal arguments about why looking at the branching Brownian motion, and not at other measure-valued branching Markov processes, like superprocesses, yields the problem closest to optimal transport. Finally, we explain how this problem can be solved numerically: the dynamical formulation of regularized unbalanced optimal transport can be discretized and solved via convex optimization.