arXiv research

Minimal surfaces' behavior at infinity resolved.

Quadratic growth of intersecting curves on surfaces resolved.

Paper proves stable minimal surfaces in 3D are flat.

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

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…

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

We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of Carnot groups. This class contains groups for which it is known that their Dehn function grows no faster than $n^2\log n$. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functio…

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Be…

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…

Optimal Liouville theorem for minimal disks in any codimension.

Constructs manifolds with infinite Betti numbers and close to quadratic volume growth.

Research shows quadratic growth in derivative maxima for certain interval diffeos with parabolic fixed points.

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

Classifies gravitational instantons with quadratic volume growth.

This article proposes a new approximation scheme for quadratic-growth BSDEs in a Markovian setting by connecting a series of semi-analytic asymptotic expansions applied to short-time intervals. Although there remains a condition which needs to be checked a posteriori, one can avoid altogether time-consuming Monte Carlo…

We show that there are topological obstructions for a noncompact manifold to admit a Riemannian metric with quadratic curvature decay and a volume growth which is slower than that of Euclidean space of the same dimension.

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.

The paper examines perimeter minimizing sets in curved spaces and finds conditions for their boundary to match a specific structure.

New minimal surfaces grow area very quickly.

Sharp time analyticity proved for heat equation on Ricci solitons.

Entire area-minimizing surfaces of density 2 are planar or quadratic

Ancient Lagrangian flows get limited convex solutions.

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

Finite intersection numbers between horizontal foliations of quadratic differentials.

We investigate a class of quadratic-exponential growth BSDEs with jumps. The quadratic structure introduced by Barrieu & El Karoui (2013) yields the universal bounds on the possible solutions. With local Lipschitz continuity and the so-called A_gamma-condition for the comparison principle to hold, we prove the existenc…

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…

Study periodic orbits in specific parts of surface moduli spaces.

Minimal area of Teichmüller curves in genus two is found to be 3π/5.

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

Paper approximates Kelly betting for wealth growth.

The paper studies ABSDEs with jumps and various growth drivers.

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

We prove that any complete metric on R^3 minus a ball with non-negative Ricci curvature and quadratic Ricci-curvature decay, has cubic volume growth.

Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.

New approach finds solutions to games with unbounded controls.

The study predicts large genus behavior of quadratic differential volumes and constants.

Paper constructs multivalued harmonic functions on R^3 using twistor methods.

We give sufficient conditions for a noncompact Riemannian manifold, which has quadratic curvature decay, to have finite topological type with ends that are cones over spherical space forms.

The paper proves unique characterization of gravitational instantons with specific volume growth.

Study shows uniqueness of solutions on complex manifolds without requiring solution decay.

We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manif…

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 …

The study connects geodesic flows on Riemann surfaces to random walks on their dual graphs.

Solves new quadratic BSDE systems for market performance analysis.

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…

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

A Liouville theorem on a specific PDE yields quadratic solutions.