arXiv research

In this paper we refine the construction and related estimates for complete Constant Mean Curvature surfaces in Euclidean three-space developed in Kapouleas (1990) by adopting the more precise and powerful version of the methodology which was developed in Kapouleas (1995). As a consequence we remove the severe restrict…

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone and 1-homogeneous curvature function. In particular this class includes the mean curvature $F=H$. We prove that a certain initial…

Classifies soap film surfaces with vertical potentials.

We consider inverse curvature flows in the $(n+1)$-dimensional Euclidean space, $n\geq 2,$ expanding by arbitrary negative powers of a 1-homogeneous, monotone curvature function $F$ with some concavity properties. We obtain asymptotical roundness, meaning that circumradius minus inradius of the flow hypersurfaces decay…

New metrics defined on SPD matrices link to divergences and curvature.

We consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ with speed given by a power of the $k$-th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume $V_{n+1}$ and the mixed volume $V_{n+1-k}$ of the evolving hypersurface. We prove that i…

DFNNs predict non-Euclidean responses from Euclidean predictors.

This paper concerns the evolution of a closed hypersurface of dimension $n(\geq 2)$ in the Euclidean space ${\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $β(\geq 1)$ of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, incl…

No compact surfaces with specific curvature can exist near singular limits.

The study characterizes round spheres in Euclidean space based on r-mean curvature conditions.

Evolving smooth, compact hypersurfaces in R^{n+1} with normal speed equal to a positive power k of the mean curvature improves a certain 'isoperimetric difference' for k >= n-1. As singularities may develop before the volume goes to zero, we develop a weak level-set formulation for such flows and show that the above mo…

Constructs hyperspheres with prescribed mean curvature in Euclidean space.

Investigates differences in solving mean curvature problems in Euclidean and Lorentz-Minkowski spaces.

Estimates statistical power for cluster analysis in biomedical research.

Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.

A new derivation is given of Branson's factorization formula for the conformally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Branson's formula from knowledge of the corresponding conformally invariant operator on Euclidean space (the k-th power…

The paper proves a theorem about mean curvature in Euclidean and hyperbolic spaces.

We give an explicit estimate of the distance of a closed, connected, oriented and immersed hypersurface of a space form to a geodesic sphere and show that the spherical closeness can be controlled by a power of an integral norm of the traceless second fundamental form, whenever the latter is sufficiently small. Further…

Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.

Radial graphs with constant mean curvature found in Euclidean space.

Unbounded convex domains have zero mean curvature on disconnected boundaries.

The study characterizes canal hypersurfaces in Euclidean spaces and their curvature properties.

Develops weak formulation for spacelike flows in pseudo-Euclidean space.

We study the use of power weighted shortest path distance functions for clustering high dimensional Euclidean data, under the assumption that the data is drawn from a collection of disjoint low dimensional manifolds. We argue, theoretically and experimentally, that this leads to higher clustering accuracy. We also pres…

No regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces are found.

Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle …

Nonparametric two sample testing is a decision theoretic problem that involves identifying differences between two random variables without making parametric assumptions about their underlying distributions. We refer to the most common settings as mean difference alternatives (MDA), for testing differences only in firs…

In this paper, we prove the concavity of $p$-entropy power of probability densities solving the $p$-heat equation on closed Riemannian manifold with nonnegative Ricci curvature. As applications, we give new proofs of $L^p$-Euclidean Nash inequality and $L^p$-Euclidean Logarithmic Sobolev inequality, moreover, an improv…

We classify all Hamiltonian stationary Lagrangian surfaces in complex Euclidean plane which are self-similar solutions of the mean curvature flow.

We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curva…

The paper classifies special solitons and shrinkers in Euclidean space.

New, algebraic surfaces found in curved spaces.

We establish the longtime existence and convergence results of the mean curvature flow of entire Lagrangian graphs in Pseudo-Euclidean space which is related to Logarithmic gradient flow.

Improved gap for mean curvature of biharmonic hypersurfaces in spheres.

I classify spacelike self-similar shrinking solutions of the mean curvature flow in pseudo-euclidean space in arbitrary codimension, if the mean curvature vector is not a null vector and the principal normal vector is parallel in the normal bundle. Moreover, I exclude the existence of such self-shrinkers in several cas…

We study ends of an oriented, immersed, non-compact, complete Willmore surfaces, which are critical points of the integral of the square of the mean curvature, in asymptotically flat spaces of any dimension; assuming the surface has $L^2$-bounded second fundamental form and satisfies a weak power growth on the area. We…

Minimal biharmonic hypersurfaces in Euclidean spaces are ideal.

We give the classification of constant mean curvature rotational surfaces of elliptic, hyperbolic, and parabolic type in the four-dimensional pseudo-Euclidean space with neutral metric.

Using certain solutions of the curve shortening flow, including self-shrinking and self-expanding curves or spirals, we construct and characterize many new examples of translating solitons for mean curvature flow in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones \cite{JLT} in dimension two. The s…

We make observations about constant mean curvature surfaces in Euclidean 3-space and their dual surfaces, and the resulting pairs of surfaces in hyperbolic 3-space under the Lawson correspondence.

The study finds larger gaps in mean curvature for biharmonic submanifolds in spheres.

Study of metrics on positive-definite matrices from power potential, linking to power means.

New method constructs surfaces with constant mean curvature.

Theory proves existence of hypersurfaces with prescribed curvature.

We use bifurcation theory to show the existence of infinite sequences isometric embeddings of tori with constant mean curvature (CMC) in Euclidean spheres that are not isometrically congruent to the CMC Clifford tori, and accumulating at some CMC Clifford torus.

The paper studies essential spectra of submanifolds in Euclidean spaces.

Estimates modes and ridges in mixed Euclidean and directional spaces.

Sharp bounds for anisotropic p-capacity of Euclidean compact sets derived using flow methods.