arXiv research

Consider a mean curvature flow of hypersurfaces in Euclidean space, that is initially graphical inside a cylinder. There exists a period of time during which the flow is graphical inside the cylinder of half the radius. Here we prove a lower bound on this period depending on the Lipschitz-constant of the initial graphi…

The paper studies a flow of spacelike curves in a Lorentz-Minkowski plane, showing convergence to a constant function.

The paper studies how certain spacelike surfaces evolve over time in a specific space.

The paper studies a flow of spacelike surfaces in Lorentz-Minkowski space, proving convergence to a hyperbolic plane.

Paper proves unique tangent flow at infinity for entropy-limited curve shortening.

The paper estimates curvature for specific hypersurfaces in a special space.

Building on a recent framework for distributionally robust optimization, we consider estimation of the inverse covariance matrix for multivariate data. We provide a novel notion of a Wasserstein ambiguity set specifically tailored to this estimation problem, leading to a tractable class of regularized estimators. Speci…

New inequality shows energy growth and decay in geometric problems.

The paper studies how spacelike surfaces evolve in Lorentz-Minkowski space over time.

The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be realized as graphical hypersurfaces in $\mathbb{R}^{n+1}$. Specifically, for an asympto…

The purpose of this paper is twofold: firstly, to establish sufficient conditions under which the mean curvature flow supported on a hypersphere with exterior Dirichlet boundary exists globally in time and converges to a minimal surface, and secondly, to illustrate the application of Killing vector fields in the preser…

Cohomology fractals illustrate complex 3-manifold properties.

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any such compact minimal surface $M$, we provide a priori radius estimate which depend…

In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname…

Positive injectivity radius for manifolds with Lie structure at infinity.

The ratio of convexity radius over injectivity radius may be made arbitrarily small within the class of compact Riemannian manifolds of any fixed dimension at least two. This is proved using Gulliver's method of constructing manifolds with focal points but no conjugate points. The approach is suggested by a characteriz…

Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.

Proves upper bound on filling radius for manifolds with positive scalar curvature.

Compact theorem for minimal surfaces with lower injectivity radius.

Injectivity radius on Stiefel manifold is π.

Lower bound on boundary injectivity radius for specific tubes.

Study gives bounds on filling radius for Riemannian manifolds.

The paper proves estimates and theorems for Kähler manifolds.

Study finds the covering radius of RM(4,8) is 26.

Clarifies definition of polarized canonical radius in Kahler Ricci flow.

The paper studies cylindrical singularities in mean curvature flow and proves their local regularity.

The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.

Upper bound on Stiefel manifold's injectivity radius found.

Upper estimate of Dirac eigenvalue linked to hyperspherical radius.

The study proves triviality and rigidity results for Ricci solitons and estimates their conjugate radius.

Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.

A translation structure equips a Riemann surface with a singular flat metric. Not much is known about the shape of a random translation surface. We compute an upper bound on the expected value of the covering radius of a translation surface in any stratum H_1(kappa). The covering radius of a translation surface is the …

This paper considers metric balls $B(p,R)$ in two dimensional Riemannian manifolds when $R$ is less than half the convexity radius. We prove that $Area(B(p,R)) \geq \frac{8}πR^2$. This inequality has long been conjectured for $R$ less than half the injectivity radius. This result also yields the upper bound $μ_2(B(p,R)…

We derive an explicit lower bound on the radius of a ball embedded in a quaternionic hyperbolic manifold.

Uniformly positive scalar curvature implies a lower bound on injectivity radius.

Paper generalizes inscribed radius estimate to hyperbolic Einstein manifolds.

We prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive

Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.

We prove that spherical spectral analysis and synthesis hold in Damek-Ricci spaces and derive two-radius theorems.

Study compares eigenvalues on spherically symmetric manifolds to Euclidean balls.

Graphical lasso may fail to fit models when data points are insufficient.

Study on rigidity of translating hypersurfaces not in graphical direction.

The paper improves bounds on injectivity radius for manifolds with positive scalar curvature.

In this note we relate the geometric notion of fill radius with the fundamental group of the manifold. We prove: ''Suppose that a closed Riemannian manifold M satisfies the property that its universal cover has bounded fill radius. Then the fundamental group of M is virtually free.'' We explain the relevance of this th…

Characterizes submanifolds with minimum ratio of diameter to focal radius.

Generalizes a soul-bound for noncompact Alexandrov spaces.

We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Moreover, we prove a generalization of Gromov's packing…

We prove a sharp estimate for the inscribed radius under certain fully nonlinear curvature flows. This estimate is asymptotically sharp on cylinders.