arXiv research

Every convex set in a generic Riemannian manifold has peculiar properties.

Survey on extending rigidity theorems to Riemannian manifolds.

Proves a higher rank rigidity theorem for convex real projective manifolds.

The paper shows that g-convex functions on manifolds are sparse.

A small projective 4-manifold created via Dehn filling.

Extends DCP framework to Hadamard manifolds for geodesically convex functions.

Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.

New method optimizes on curved manifolds without curvature dependence.

Unique compact Fuchsian manifolds with convex boundary are determined by their boundary.

Study shows infimum of dual volume equals convex core volume for hyperbolic 3-manifolds.

New method for optimization on Hadamard manifolds with curvature-independent guarantees.

We study a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul which asserts that for a compact manifold without boundary the holonomies of properly convex structures form an …

Let K be a connected Lie group and M a Hamiltonian K-manifold. In this paper, we introduce the notion of convexity of M. It implies that the momentum image is convex, the moment map has connected fibers, and the total moment map is open onto its image. Conversely, the three properties above imply convexity. We show tha…

A new method solves convex optimization problems on manifolds efficiently.

Research shows how certain flat structures behave in specific convex domains.

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

Entropy rigidity proven for 3D and higher convex projective manifolds.

Geodesic flow mixing on convex projective manifolds proven.

3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.

Properly convex manifolds with generalized cusps have irreducible holonomy.

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…

Contact manifolds' momentum polytopes are convex.

Unique hyperbolic manifolds identified by boundary pleating.

Convex hypersurfaces in curved spaces bound convex regions.

A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset $S$ in the cases where the boundary of $S$ contains a geodesic segment, the boundary of $S$ is $C^2$ o…

This paper extends a 3D result to higher dimensions for manifolds with positive curvature.

3-manifolds with convex boundary are rigid in certain curvature conditions.

We show that a non-compact (forward) complete Finsler manifold whose Holmes- Thompson volume is infinite admits no non-trivial convex functions. We apply this result to some Finsler manifolds whose Busemann function is convex.

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $Σ$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric properties of bent manifolds and algebraic properties of their fundamental groups. We th…

The study proves conditions for convex hypersurfaces in Riemannian manifolds to be rational homology spheres.

Generalizes Giroux's result to higher dimensions and applies to contact manifolds.

Proof that convex structures on manifolds are open and closed.

The main goal of this paper is to present results of existence and non-existence of convex functions on Riemannian manifolds and, in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove that the conservativity of the geodesic flow on a Rieman- nain manifold with in…

The study proves Liouville theorems on curved manifolds with convex boundaries.

The usual notion of set-convexity, valid in the classical Euclidean context, metamorphoses into several distinct convexity types in the more general Riemannian setting. By studying this phenomenon in reverse, we characterize complete manifolds for which certain convexity types are assumed a priori to coincide.

Study shows curvature bounds for convex hypersurfaces in specific manifolds.

Plumbing of surfaces embeds in hyperbolic 4-manifolds.

The paper connects convex functions to p-subharmonic functions and proves their equivalence.

Lower bound found for Steklov eigenvalue on curved manifolds.

We study the topology of complete Finsler manifolds admitting convex functions

Geodesically convex functions are continuous on Riemannian manifolds.

We prove the existence of free boundary minimal annuli inside suitably convex subsets of three-dimensional Riemannian manifolds with nonnegative Ricci curvature $-$ including strictly convex domains of the Euclidean space $\mathbb{R}^3$.

Given a Riemannian manifold M and a hypersurface H in M, it is well known that infinitesimal convexity on a neighborhood of a point in H implies local convexity. We show in this note that the same result holds in a semi-Riemannian manifold. We make some remarks for the case when only timelike, null or spacelike geodesi…

New theorem on 3-manifolds with curvature and convex boundary.

Local invertibility of ray transforms on convex manifolds.

Study of manifolds with specific curvature properties using capillary surfaces.

The null energy condition is characterized via convexity of entropy in Lorentzian manifolds.

Locally convex compact immersed hypersurfaces in Finsler-Hadamard manifolds with bounded T-curvature are considered. We prove that such hypersurfaces are embedded as the boundary of convex body under certain conditions on the normal curvatures