arXiv research

New minimal surfaces grow area very quickly.

Minimal surfaces' behavior at infinity resolved.

Estimates area and spectrum of stable minimal surfaces in Euclidean and hyperbolic spaces.

We prove flatness of complete Riemannian planes and cylinders without conjugate points under optimal conditions on the area growth.

In this paper, we prove uniform lower bounds on the volume growth of balls in the universal covers of Riemannian surfaces and graphs. More precisely, there exists a constant $δ>0$ such that if $(M,hyp)$ is a closed hyperbolic surface and $h$ another metric on $M$ with $\area(M,h)\leq δ\area(M,hyp)$ then for every radiu…

Study proves Chern-Osserman type equality for surfaces in Euclidean space.

Classifies area-minimizing surfaces in R^4 as algebraic.

We establish parabolicity and quadratic area growth for minimal surfaces-with-boundary contained in regions of R^3 which are within a sub-logarithmic factor of the exterior of a cone. Unlike previous work showing that these two properties hold for minimal surfaces-with-boundary contained between two catenoids, we do no…

Paper proves stable minimal surfaces in 3D are flat.

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…

Random surfaces with long systoles created from graph theory ideas.

Study finds all local models for flat metrics with isolated singularities.

Study counts minimal Lagrangians in hyperbolic surfaces with precise growth rate.

Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.

In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern-Osserman inequality. In th…

Quadratic growth of intersecting curves on surfaces resolved.

We study filling sets of simple closed curves on punctured surfaces. In particular we study lower bounds on the cardinality of sets of curves that fill and that pairwise intersect at most k times on surfaces with given genus and number of punctures. We are able to establish orders of growth for even k and show that for…

The study counts arcs on hyperbolic surfaces, providing asymptotic growth formulas.

We are interested in the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g,n)$. This maximum is shown to be strictly increasing in terms of the number of cusps for small values of $n$. We also show that this function is greater than a function that…

Minimal surfaces with bounded index have linearly growing genus.

The paper studies stability and area growth of $λ$-hypersurfaces.

Inspired by an argument of Ros [15] -- we use the López-Ros deformation to give another proof of the fact -- due to Meeks and Wolf [13] -- that the only smooth, connected, singly-periodic minimal surfaces in $\Real^3$ with the area growth of two planes are the singly-periodic Scherk surfaces.

Optimal Liouville theorem for minimal disks in any codimension.

This paper studies the critical dynamics of random surfaces, focusing on area and genus evolution.

We prove that a connected properly immersed minimal surface in Euclidean 3-space with infinite symmetry group whose intersection with a ball of radius R is less than 2\piR^2 is a plane, a catenoid or a Scherk singly-periodic minimal surface. In particular, we prove that the only periodic minimal desingularization of a …

Paper proves compactness and rigidity of $λ$-surfaces in 3D.

We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.

Study on $k$-surfaces in negatively curved 3-manifolds, focusing on energy and entropy.

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…

The study finds a formula for the number of geodesic surfaces in a specific 3-manifold.

Researchers prove existence and uniqueness of translators for mean curvature flow.

Optimal inequalities for metric surfaces derived from filling minimality.

Study proves no minimal surfaces can be contained in certain half-spaces or cones.

Minimal Gaussian surfaces partitioning space with minimum area.

Paper studies stability of curved surfaces in a half-space.

In this paper, we introduce a definition of $λ$-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that $λ$-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $λ$-hypersurfaces with …

Macroeconomic theories of growth and wealth distribution have an outsized influence on national and international social and economic policies. Yet, due to a relative lack of reliable, system wide data, many such theories remain, at best, unvalidated and, at worst, misleading. In this paper, we introduce a novel econom…

Improved flatness in annuli using PDE methods.

An Abelian differential gives rise to a flat structure (translation surface) on the underlying Riemann surface. In some directions the directional flow on the flat surface may contain a periodic region that is made up of maximal cylinders filled by parallel geodesics of the same length. The growth rate of the number of…

Study finds minimum growth rate for surface solutions.

Directly proves logarithmic systolic growth for all hyperbolic surfaces.

Entropy defined for submanifolds; applies to mean curvature flow limits of surfaces.

The study improves the upper bound for the first eigenvalue of Laplacian on compact surfaces of large genus.

Model predicts growth competition on curved surfaces.

Unified theory explains housing cycle across metros, showing credit expansion impacts.

The paper extends rigidity results for $λ$-self-expanders to hyperplanes, spheres, and cylinders.

It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc. We prove that in genus two …

Study growth rates of harmonic functions on curved surfaces.