arXiv research

The study finds billiard trajectories with infinitely many reflections in certain cones.

This paper proves integrability of Birkhoff billiards inside convex cones.

The paper examines conditions for stochastic invariance of cones in SPDEs with jumps.

In this paper we study the area minimizing problem in some kinds of conformal cones. This concept is a generalization of the cones in Eulcidean spaces and the cylinders in product manifolds. We define a non-closed-minimal (NCM) condition for bounded domains. Under this assumption and other necessary conditions we estab…

The Lee-Gauduchon cone is a convex cone of cohomology classes for complex manifolds.

The study of limit cones for multi-Fuchsian representations in $(\mathrm{PSL}_2\mathbf{R})^d$.

In order to obtain a closed orientable convex projective four-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic four-manifold through a continuous path of projective cone-manifolds.

Anti-de Sitter spacetimes embed cone-metrics as bent Cauchy surfaces.

Proves stability of cone-volume measure with nearly constant density.

We consider graphical solutions to mean curvature flow and obtain a stability result for homothetically expanding solutions coming out of cones of positive mean curvature: If another solution is initially close to the cone at infinity, then the difference to the homothetically expanding solution becomes small for large…

We characterize embedded $\C^1$ hypersurfaces of $\R^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most $m<3/2$. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform r…

Study geodesic flows on cones over Riemannian manifolds, showing superintegrability.

We prove an existence theorem for convex hypersurfaces of prescribed Gauss curvature in the complement of a compact set in Euclidean space which are close to a cone.

Study proves radial symmetry in convex cones using subharmonic functions.

Paper constructs geometric transitions between hyperbolic and Anti-de Sitter geometries.

CoNES optimizes blackbox functions using convex optimization and information geometry.

In this paper, we propose a method for image-set classification based on convex cone models, focusing on the effectiveness of convolutional neural network (CNN) features as inputs. CNN features have non-negative values when using the rectified linear unit as an activation function. This naturally leads us to model a se…

Closed hyperbolic manifolds are proven to minimize volume over all Alexandrov spaces with curvature bounded below by -1 in the same bilipschitz class. As a corollary compact convex cores with totally geodesic boundary are proven to minimize volume over all hyperbolic manifolds in the same bilipschitz class. Also, close…

Study coning totally geodesic boundaries of hyperbolic manifolds.

Inverse curvature flows shape star-shaped hypersurfaces into spheres.

Study on $p$-Laplace equation in convex cones, proving rigidity under specific conditions.

The paper provides uniform length estimates for trajectories on flat cone surfaces.

Fast algorithm recovers principal eigenvector from noisy matrices.

Smooths out complex shapes into simpler forms.

Study on Serrin's problem in convex cones with rigidity results and geometric inequalities.

Paper solves a complex equation for unbounded convex sets.

We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on selfdual cones, Weyl groups of certain Kac-Moody algebras …

Study on extremals in sub-Lorentzian geometry defined by antinorm.

Compact hypersurfaces minimize area in convex cones with free boundary.

Clarifies definitions of global hyperbolicity in various spaces.

We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some …

Given a convex cone in the \emph{prescribed} warped product, we consider hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve along the inverse mean curvature flow, then, by using the convexity of …

dboost optimizes prediction models for convex cone problems.

The paper classifies a space of generalized cusps and its moduli.

We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variation…

Learning graph representations via low-dimensional embeddings that preserve relevant network properties is an important class of problems in machine learning. We here present a novel method to embed directed acyclic graphs. Following prior work, we first advocate for using hyperbolic spaces which provably model tree-li…

Study shows unique tangent cones for area-minimizing currents at boundary points.

If a convex body C has modular and irreducible face lattice (and is not strictly convex), there is a face-preserving homeomorphism from C to a section of a cone of hermitian matrices or C has dimension 8, 14 or 26.

New conical metrics found on toric varieties with convex cones.

Open, connected, saturated sets W without holonomy in codimension one foliations play key roles as fundamental building blocks. Here, for the case of foliated 3-manifolds, we produce a finite system of closed, convex, non-overlapping polyhedral cones in the first cohomology of W with real coefficients such that the iso…

We prove that every closed oriented 3-manifold admits a hyperbolic cone-manifold structure with cone-angle arbitrarily close to 2pi.

Study convex hyperbolic cone-metrics on 3-manifold boundaries, proving unique bent realizations.

An affine manifold is a manifold with an affine structure, i.e. a torsion-free flat affine connection. We show that the universal cover of a closed affine 3-manifold $M$ with holonomy group of shrinkable dimension (or discompacité in French) less than or equal to two is diffeomorphic to $\bR^3$. Hence, $M$ is irreducib…

We prove curvature estimates for general curvature functions. As an application we show the existence of closed, strictly convex hypersurfaces with prescribed curvature $F$, where the defining cone of $F$ is $\C_+$. $F$ is only assumed to be monotone, symmetric, homogeneous of degree 1, concave and of class $C^{m,\al}$…

We consider strictly convex hypersurfaces with the boundary which meets a strictly convex cone perpendicularly. We prove that if these hypersurfaces expand inside this cone, driven by the power of the Gauss curvature, then the evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece…

We show that the cone associated with a moment map for an action of a torus on a contact compact connected manifold is a convex polyhedral cone and that the moment map has connected fibers provided the dimension of the torus is bigger than 2 and that no orbit is tangent to the contact distribution. This may be consider…

Affine deformations of convex cones on projective surfaces.

Establishes convexity and coercivity of K-energy functional for complex tori.