arXiv research

Milken revolutionized finance with junk bonds, boosting private equity and tech growth.

Alternative proof of Michael-Simon-Sobolev inequality using optimal transport.

Optimizes transport on submanifolds for curvature inequalities.

Proves inequalities for tensor fields on submanifolds using ABP method.

Proves inequality for tensor fields on curved spaces.

Michael-Simon inequality proven for anisotropic energies close to area.

The paper proves Michael-Simon inequalities in hyperbolic space using novel curvature flows.

This paper continues the study of Alexandrov-Fenchel inequalities for quermassintegrals for $k$-convex domains. It focuses on the application to the Michael-Simon type inequalities for $k$-curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a do…

Sharp Sobolev and Michael-Simon inequalities on curved manifolds.

Explains geometric inequalities for minimal hypersurfaces.

Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.

Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.

New inequalities derived for hyperbolic space via specific flows.

Proves new Sobolev inequalities for submanifolds in manifolds with nonnegative intermediate Ricci curvature.

Study examines surfaces with bounded fractional mean curvature, proving control over local parametrization.

The paper is devoted to generalization of well-known Michael's Selection theorem on the case of extension dimension.

The study proves inequalities on curved spaces without global curvature bounds.

The paper proves new inequalities and flow properties for hypersurfaces.

We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.

Building on the work of and answering a question by Michael Harrison, we show that any contact structure on Euclidean 3-space induced by a line fibration is diffeomorphic to the standard contact structure.

Completeness of surface metrics established for Sobolev spaces.

The biharmonic flow and Willmore flow are studied in higher dimensions using geometric evolution equations.

Let G be a k-step Carnot group. We prove an isoperimetric-type inequality for compact C^2-smooth immersed hypersurfaces with boundary, involving the horizontal mean curvature of the hypersurface. This generalizes an inequality due to Michael and Simon, and Allard, independently. Some applications are discussed.

Johnson, Kidwell, and Michael showed that intrinsically knotted graphs have at least 21 edges. Also it is known that K7 and the thirteen graphs obtained from K7 by rY moves are intrinsically knotted graphs with 21 edges. We prove that these 14 graphs are the only intrinsically knotted graphs with 21 edges.

In joint work with Yang Huang, we defined a canonical absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields. A similar grading was defined on embedded contact homology by Michael Hutchings. In this paper we show that the isomorphism between these homology theories defined by Colin-G…

Various Seiberg-Witten Floer cohomologies are defined for a closed, oriented 3-manifold; and if it is the mapping torus of an area-preserving surface automorphism, it has an associated periodic Floer homology as defined by Michael Hutchings. We construct an isomorphism between a certain version of Seiberg-Witten Floer …

New Sobolev inequality found for mean convex spacelike submanifolds in Minkowski space.

The aim of this paper is to prove isoperimetric inequalities on submanifolds of the Euclidean space using mass transportation methods. We obtain a sharp ?weighted isoperimetric inequality? and a nonsharp classical inequality similar to the one obtained by J. Michael and L. Simon. The proof relies on the description of …

Paper doubles Hessian estimates for special Lagrangian equation with constraints.

Proves Hessian estimates for special Lagrangian equation with new proofs.

We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as later results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation a mean value theorem with respect to `heat spheres' is proved for heat equation …

Paper tackles circularity issues in machine learning predictions.

Sharp inequality for submanifolds in curved spaces.

Survey of geometry developments, including complex structures on surfaces.

This paper contains a construction of a finite set X in the boundary of the unit 3-ball in R^3 whose minimal tree is knotted. The example answers Problem 5.17 in ''Problems in Low-dimensional Topology'' by Rob Kirby posed by Michael Freedman: ''Given a finite set of points X in the boundary of B^3, let T be a tree in B…

In this paper, we study the regularity of asymptotically hyperbolic metrics with Einstein condition near boundary and Weyl curvature smooth enough in arbitrary dimension. Following Michael Anderson's method, we show that $C^{m,α}$ conformally compact Riemannian metrics with Einstein equation vanishing to finite order n…

We review some approaches to the understanding of fluctuations in some models used to describe socio and economic systems. Our approach builds on the development of a simple Langevin equation that characterises stochastic processes. This provides a unifying approach that allows first a straightforward description of th…

Some of the most known integral inequalities are the Sobolev, Hardy and Rellich inequalities in Euclidean spaces. In the context of submanifolds, the Sobolev inequality was proved by Michael-Simon and Hoffman-Spruck. Since then, a sort of applications to the submanifold theory has been derived from those inequalities. …

We study boundary regularity for conformally compact Einstein metrics in even dimensions by generalizing the ideas of Michael Anderson. Our method of approach is to view the vanishing of the Ambient Obstruction tensor as an nth order system of equations for the components of a compactification of the given metric. This…

Given a configuration $\mathbf{x}$ of $n$ distinct points in hyperbolic $3$-space $H^3$, Michael Atiyah associated $n$ polynomials $p_1,\ldots,p_n$ of a variable $t \in \mathbb{C}P^1$, of degree $n-1$, and conjectured that they are linearly independent over $\mathbb{C}$, no matter which configuration $\mathbf{x}$ one s…

In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle, and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by t…

Recently, John Franks and Michael Handel proved that, for $g\geq 3$ and $n\leq 2g-4$, every homomorphism from the mapping class group of an orientable surface of genus $g$ to $\GL (n,\C)$ is trivial. We extend this result to $n\leq 2g-1$, also covering the case $g=2$. As an application, we prove the corresponding resul…

Two-dimensional Lagrangian mean curvature equation solved with new inequality.

Topological complexity for closed 1-forms

In this note we generalize an extension theorem in [5] and [9] of the mean curvature flow to the H^{k} mean curvature flow under some extra conditions. The main difficult problem in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the H^{k} mean curvature flow, and to do a sui…

This is a companion paper of arXiv:1601.03586. We study Coulomb branches of unframed and framed quiver gauge theories of type $ADE$. In the unframed case they are isomorphic to the moduli space of based rational maps from ${\mathbb C}P^1$ to the flag variety. In the framed case they are slices in the affine Grassmannia…

Let $Σ$ be a surface with a symplectic form, let $φ$ be a symplectomorphism of $Σ$, and let $Y$ be the mapping torus of $φ$. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in $\R\times Y$, with cylindrical ends asymptotic to periodic orbits of $φ$ or multiple covers thereof, are bound…

We study warped compactifications of string/M theory with the help of effective potentials, continuing previous work of the last two authors and Michael R. Douglas presented in arXiv:1206.1885. The dynamics of the conformal factor of the internal metric, which is responsible for instabilities in these constructions, is…