arXiv research

Sharp estimate for flow in any dimension.

Sharp area estimates for minimal submanifolds in curved spaces.

Estimate sphere area in Sol group up to a factor of 10.

We estimate whether there is an embedding from one n-dimensional rectangle into another which expands every k-dimensional area. Our estimate is sharp up to a constant factor in each dimension.

We obtain area growth estimates for constant mean curvature graphs in $\mathbb{E}(κ,τ)$-spaces with $κ\leq 0$, by finding sharp upper bounds for the volume of geodesic balls in $\mathbb{E}(κ,τ)$. We focus on complete graphs and graphs with zero boundary values. For instance, we prove that entire graphs in $\mathbb{E}(κ…

The paper studies volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.

Minimal surfaces in hyperbolic space have a sharp area bound.

In this article, we investigate the geometry of critical metrics of the volume functional on an $n$-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifol…

Sharp upper bound for minimal graph area in unit ball established.

Sharp lower bound found for area of vector fields on spherical annuli.

The paper studies minimal surfaces in 3D spheres and balls, confirming conjectures and identifying new surfaces.

New examples show flat singular sets can be arbitrarily complex.

Sharp bounds found on shortest geodesic on punctured spheres.

Minimal surfaces in a ball have limited area.

This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to expl…

We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…

We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the cor…

We prove a sharp estimate on the expected value of the integral of the index of a simple random walk on the square or triangular lattice. This gives new lower bounds on the averaged Dehn function, which measures the expected area needed to fill a random curve with a disc.

The paper develops new methods to study sharp isoperimetric properties on complex spaces.

Estimates Bartnik mass for metrics with nonnegative Gauss curvature.

The paper establishes inequalities for convex curves and applies them to lattice point estimates.

Sharp lower bound found for integral varifolds' mean curvature.

Affine $λ$-equidistants of convex polygons with parallel opposite sides have applications to isoperimetric inequalities.

Extends width estimates to family case using index theory.

Study flat flow solutions to Mullins-Sekerka and area-preserving curvature flows on planar flat torus.

Sharp inequalities for curved surfaces and cones.

Using a new method we give elementary estimates for the capacity of non-contractible annuli on cylinders and provide examples, where these inequalities are sharp. Here the lower bound depends only on the area of the annulus. In the case of constant curvature this lower bound is obtained with the help of a symmetrizatio…

Sharp Minkowski inequality for convex surfaces in curved spaces.

Study characterizes compact Einstein-type manifolds with boundary.

Suppose $C$ is a compact, $n$-edged two-cell of the centered dual decomposition of a locally finite set in the hyperbolic plane, a coarsening of the Delaunay tessellation which was introduced in the author's prior work. We describe an effectively computable lower bound on the area of $C$, given an $n$-tuple of positive…

Establish optimal Lipschitz lower bounds for functions on manifolds with negative curvature, revealing interplay between width, boundary area, and topology.

Improved mass-capacity bounds for specific 3D manifolds.

We give a sharp upper bound for the area of a minimal two-sphere in a three-manifold (M,g) with positive scalar curvature. If equality holds, we show that the universal cover of (M,g) is isometric to a cylinder.

New results show area-minimizing surfaces have fewer singularities than expected.

In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or equal to 6 and is not round must have an embedded minimal sphere of area strictly smaller than $4π$ and index at most one…

Constructs area-minimizing submanifolds with fractal singularities.

Sharp inequality for Lorentzian spaces with timelike Ricci bounds.

We extend to higher dimensions earlier sharp bounds for the area of two dimensional free boundary minimal surfaces contained in a geodesic ball of the round sphere. This follows work of Brendle and Fraser-Schoen in the euclidean case.

New method to bound Laplacian eigenvalues of geodesic balls.

Study area-minimizing hypersurfaces in manifolds with controlled curvature.

For any given natural number $k$, this paper gives upper bounds on the radius of a packing of a complete hyperbolic surface of finite area by $k$ equal-radius disks in terms of the surface's topology. We show that the bounds given here are sharp in some cases and not sharp in others.

Let Σbe a k-dimensional minimal surface in the unit ball B^n which meets the unit sphere orthogonally. We show that the area of Σis bounded from below by the volume of the unit ball in R^k. This answers a question posed by R. Schoen.

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. R…

We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly increase the first eigenvalue normalized by area if the initial su…

Study area-minimizing hypersurfaces in singular manifolds with nonnegative scalar curvature.

Sharp generalization of boundary regularity for area minimizing currents with arbitrary multiplicity.

We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such …

Minimal surfaces and average area ratio found to be maximized by hyperbolic metrics.