arXiv research

New formulas limit minimal submanifolds' area in curved spaces.

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 the formation of singularities for the mean curvature flow of monotone Lagrangians in $\C^n$. More precisely, we show that if singularities happen before a critical time then the tangent flow can be decomposed into a finite union of area-minimizing Lagrangian cones (Slag cones). When $n=2$, we can improve this…

The paper evaluates the importance of monotonicity in AI fairness across various fields.

Study on singularities in area-minimizing currents, proving unique tangent cones and rectifiability.

The paper proves weighted monotonicity theorems in various spaces and applies them to minimal surfaces.

The paper proves formulas for capillary surfaces and applies them to inequalities and area estimates.

RG-2 flow preserves area monotonicity in asymptotically flat spacetimes.

Derives monotonic quantities for $p$-harmonic functions on manifolds.

Motivated by recent work of Choquet-Bruhat, Chrusciel, and Martin-Garcia, we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volu…

Study optimal portfolio allocation in a general semimartingale model.

Solves a long-standing convex geometry problem about mixed volumes.

In this note we prove a simple relation between the mean curvature form, symplectic area, and the Maslov class of a Lagrangian immersion in a Kähler-Einstein manifold. An immediate consequence is that in Kähler-Einstein manifolds with positive scalar curvature, minimal Lagrangian immersions are monotone.

Study on relative entropy for hypersurfaces in hyperbolic space.

In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ geominimal surface area is motivated by recent work on the extension of the $L_p$ a…

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

New theory for area of Legendrian surfaces, proving smoothness and variational results.

In this sequel paper we give a shorter, second proof of the monotonicity of the Hawking mass for time flat surfaces under spacelike uniformly area expanding flows in spacetimes that satisfy the dominant energy condition. We also include a third proof which builds on a known formula and describe a class of sufficient co…

Improved mass-capacity bounds for specific 3D manifolds.

New quantity helps map homotopy classes in complex spaces.

Study eigenvalues of surfaces with collapsing handles or cross caps.

In this article, we introduce a new type of mean curvature flow for bounded star-shaped domains in space forms and prove its longtime existence, exponential convergence without any curvature assumption. Along this flow, the enclosed volume is a constant and the surface area evolves monotonically. Moreover, for a bounde…

Unique tangent cones found for boundary points of 2D almost-minimizing currents.

The paper derives inequalities for $p$-capacitary functions in 3-manifolds with nonnegative scalar curvature.

Proves flows of two-convex Lagrangians are regular, global, and converge.

Study on minimal surfaces with constraints on index and branching order.

We obtain a Chern-Osserman type equality of a complete properly immersed surface in Euclidean space, provided the L^2-norm of the second fundamental form is finite. Also, by using a monotonicity formula, we prove that if the L^2-norm of mean curvature of a noncompact surface is finite, then it has at least quadratic ar…

Notwithstanding almost forty years of efforts, the market for paintings still lacks a widely accepted price index. In this paper, we introduce a simple and intuitive metric to construct such index. Our metric is based on the price of a painting divided by its area. This formulation rests on a solid mathematical foundat…

We analyze the signature type of a cascade of periodic orbits associated to period doubling renormalizable maps of the two dimensional disk. The signature is a sequence of rational numbers which describes how periodic orbits turn each other and is invariant by topological conjugacies that preserve orientation. We prove…

In this article, we continue the work in \cite{GL} and study a normalized hypersurface flow in the more general ambient setting of warped product spaces. This flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. As an application, the i…

The Orlicz-Brunn-Minkowski theory receives considerable attention recently, and many results in the $L_p$-Brunn-Minkowski theory have been extended to their Orlicz counterparts. The aim of this paper is to develop Orlicz $L_φ$ affine and geominimal surface areas for single convex body as well as for multiple convex bod…

The consultative papers for the Basel II Accord require rating systems to provide a ranking of obligors in the sense that the rating categories indicate the creditworthiness in terms of default probabilities. As a consequence, the default probabilities ought to present a monotonous function of the ordered rating catego…

Study spherical convex bodies using $L_p$-floating areas and curvature entropy.

New method solves quantile crossing problem in econometrics.

A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if its vertices lie on a metric circle, horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Convex such $n$-gons are parametrized by the subspaces of $(0,\infty)^n$ that contain their side length collections,…

New pseudometrics defined on knot spaces based on curve thickness and length.

The paper shows how Yang-Mills-Higgs energies converge to the $(n-2)$-area functional.

Michael-Simon inequality proven for anisotropic energies close to area.

We prove that capillary surfaces converge to a specific energy density as the angle approaches zero.

Analyzes branch points of area-minimizing currents with non-2 planar frequency.

We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed convex hypersurface converges to a round sphere. The proof is based on the mono…

We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small open subset of the sphere equals to the Calabi invariant of the diffeomorphism. Thi…

Given a closed, oriented surface, possibly with boundary, and a mapping class, we obtain sharp lower bounds on the number of fixed points of a surface symplectomorphism (i.e. area-preserving map) in the given mapping class, both with and without nondegeneracy assumptions on the fixed points. This generalizes the Poinca…

Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $Ω$ with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geome…

Minimal surfaces in CAT(0) spaces are well-behaved except at a few points.

In this article, we propose the notion of the general $p$-affine capacity and prove some basic properties for the general $p$-affine capacity, such as affine invariance and monotonicity. The newly proposed general $p$-affine capacity is compared with several classical geometric quantities, e.g., the volume, the $p$-var…

The doubling conjecture for positive scalar curvature is proven under certain conditions.

A new method for disentangled latent spaces in VAEs that can manipulate attributes.