arXiv research

Researchers prove rigidity of convex polytopes in hyperbolic space using spinor techniques.

The study proves a rigidity theorem for convex domains in hyperbolic spaces.

We establish a microscopic convexity principle for nonlinear elliptic and parabolic partial differential equations in general form.

Proves monotonicity of parabolic frequency on all manifolds without curvature assumptions.

Archimedes determined the center of gravity of a parabolic section as follows. For a parabolic section between a parabola and any chord $AB$ on the parabola, let us denote by $P$ the point on the parabola where the tangent is parallel to $AB$ and by $V$ the point where the line through $P$ parallel to the axis of the p…

The study proves splitting theorems for manifolds with specific curvature and boundary conditions.

Abstract: Proves generic torus diffeomorphisms act parabolically and non-properly on fine curve graph and have generalized rotation sets.

We study polar orbitopes, i.e. convex hulls of orbits of a polar representation of a compact Lie group. The face structure is studied by means of the gradient momentum map and it is shown that every face is exposed and is again a polar orbitope. Up to conjugation the faces are completely determined by the momentum poly…

New findings on domains without parabolic minimal submanifolds and weakly hyperbolic domains.

This paper studies the parabolic free boundary problem arising from pricing American-style put options on an asset whose index follows a geometric Brownian motion process. The contribution is to propose a condition for that the early exercise boundary is a convex function.

Using the theory of Weyl structures, we give a natural generalization of the notion of essential conformal structures and conformal Killing fields to arbitrary parabolic geometries. We show that a parabolic structure is inessential whenever the automorphism group acts properly on the base space. As a corollary of the g…

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural ass…

Paper solves Musielak-Orlicz-Gauss image problem using parabolic flows.

We study convexity and monotonicity properties of option prices in a model with jumps using the fact that these prices satisfy certain parabolic integro-differential equations. Conditions are provided under which preservation of convexity holds, i.e. under which the value, calculated under a chosen martingale measure, …

The paper proves smoothness of transition layers in the Allen-Cahn equation.

In this essay, we study the sufficient and necessary conditions for a Randers metrc to be of constant Ricci curvature without the restriction of strong convexity (regularity). The classification result for the case $\|β\|_α>1$ is provided, which is similar to the famous Bao-Robles-Shen's result for strongly convex Rand…

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is pro…

In this paper, we study the structure of the pointed-Gromov-Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger-Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any Ricci shrinker limit space is co…

In this paper, a class of fully nonlinear flows with nonlinear Neumann type boundary condition is considered. This problem was solved partly by the first author under the assumption that the flow is the parabolic type special Lagrangian equation in $\mathbb{R}^{2n}$. We show that the convexity is preserved for solution…

Proves planarity and convexity for ancient solutions of mean curvature flow.

The paper proves an inequality and describes a curve flow in centro-affine geometry.

Given a parabolic geometry on a smooth manifold $M$, we study a natural affine bundle $A \to M$, whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic geometry defines a reductive Cartan geometry on $A$, which induces an almost bi-Lagrangian structure on $A$ a…

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of curvature is a second order system of partial differential equations which we sho…

Anisotropic obstacle problems and Stefan problem studied with evolving surfaces.

A major challenge in current optimization research for deep learning is to automatically find optimal step sizes for each update step. The optimal step size is closely related to the shape of the loss in the update step direction. However, this shape has not yet been examined in detail. This work shows empirically that…

Solves curvature equations using parabolic flows in various spaces.

This article follow the article {http://hal.archives-ouvertes.fr/hal-00361030/fr/} in which the author characterize the fact of being of finite volume for a convex projective surface. We show here that the moduli space $β_f(Σ_{g,p})$ of the convex projective structure on the surface $Σ_{g,p}$ of genius $g$ with $p$ pun…

The paper studies how surfaces evolve in a cone under a specific flow.

We prove that stability -- a strong quasiconvexity property -- pulls back under proper actions on proper metric spaces. This result has several applications, including that convex cocompact subgroups of both mapping class groups and outer automorphism groups of free groups are stable. We also characterize stability in …

We study properly immersed ancient solutions of the codimension one mean curvature flow in $n$-dimensional Euclidean space, and classify the convex hulls of the subsets of space reached by any such flow. In particular, it follows that any compact convex ancient mean curvature flow can only have a slab, a halfspace or a…

The paper proves inequalities for convex capillary hypersurfaces in a half-space.

Reduces conjecture for Artin groups to simpler cases.

New Harnack inequality for curve shortening flow without convexity.

We prove longtime existence and estimates for solutions to a fully nonlinear Lagrangian parabolic equation with locally $C^{1,1}$ initial data $u_0$ satisfying either (1) $-(1+η) I_n\leq D^2u_0 \leq (1+η)I_n$ for some positive dimensional constant $η$, (2) $u_0$ is weakly convex everywhere or (3) $u_0$ satisfies a larg…

In this paper we study a flow by minkowskian curvature where we have a different Minkowski plane at each time. We derive some evolution formulas, present sufficient hypotesis for the short time existence and convexity of solutions and study the motion considering a particular type of families of Minkowski norms. Also, …

Affine deformations of convex cones on projective surfaces.

In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any …

Study flute surfaces and Loch Ness monster, proving their parabolicity and uniformization.

Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply convergence of convex hypersurfaces to spheres under these flows, improving earl…

Defines connections on parabolic vector bundles for Lie algebroids.

In this paper, we study the bound states of quantum layers. We prove that for the quantum layer built over a parabolic manifold which is not totally geodesic, if the second fundamantal form decays sufficiently fast, then the bound states exist. In the 2d case, we prove that the quantum layer over a convex surface whose…

Criterion found for Lie algebroid connections on parabolic bundles.

Study of spacelike discs in Minkowski cones, proving self-similar expansion.

Computes deformations of parabolic structures on Riemann surfaces.

Paper proves rigidity for self-similar solutions in 3D flows.

Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an elementary approach to the geometry of parabolic subalgebras, over an arbitrary fiel…

Study strip deformations of hyperbolic polygons with decorated vertices.

We provide some criteria to $p$-parabolicity of Riemannian submersions. In particular, if $N$ is $p$-parabolic and $π:M\to N$ is a Riemannian submersion with uniformly bounded volume of fibers, then $M$ is also $p$-parabolic. In the case of warped manifolds we characterize $p$-parabolicity in terms of a volume growth c…