arXiv research

New weighted surface area measures for convex bodies with applications.

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

New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.

Introduces new weighted floating functions and affine surface areas.

Study on $L_p$ affine surface areas and their inequalities for convex bodies.

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

We use a new approach that we call unification to prove that standard weighted double bubbles in $n$-dimensional Euclidean space minimize immiscible fluid surface energy, that is, surface area weighted by constants. The result is new for weighted area, and also gives the simplest known proof to date of the (unit weight…

Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.

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

In this paper we study the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, $λ_1$-weighted surface area, and $λ_2$-weighted volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our main result is a complete classification of all …

Entropy study of geodesic flow on convex projective surfaces.

The paper defines surface area for graphs and derives spectral estimates.

We prove two weighted geometric inequalities that hold for strictly mean convex and star-shaped hypersurfaces in Euclidean space. The first one involves the weighted area and the area of the hypersurface and also the volume of the region enclosed by the hypersurface. The second one involves the total weighted mean curv…

Hyperplanes, hyperspheres and hypercylinders in $\Bbb R^n$ with suitable densities are proved to be weighted minimizing by a calibration argument. Also calibration method is used to prove a weighted minimal hypersurface is weighted area-minimizing locally.

The paper studies hanging chains and surfaces in degenerate geometries.

The paper proves Calabi-Bernstein type results for minimal and maximal surfaces in 3D and 3D-L spacetime.

The study connects minimal and maximal surfaces in 3D and 3-L space.

For non-smooth surfaces, the measure of Brownian loops is derived using the Polyakov-Alvarez formula.

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

Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in $\p…

Consider a closed marked flat surface $S$ of genus $g\geq 2$ and area 1 and its universal covering $\tilde{S}$. We show that the measure class of the Hausdorff measure of the Gromov boundary of $\tilde{S}$ uniquely determines $S$.

We revisit the contact measures introduced by Firey, and further developed by Schneider and Teufel, from the perspective of the theory of valuations on manifolds. This reveals a link between the kinematic formulas for area measures studied by Wannerer and the integral geometry of curved isotropic spaces. As an applicat…

Given a measured lamination on a finite area hyperbolic surface we consider a natural measure Mon the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure M gives summation id…

Study shows limits of Fuchsian surfaces in hyperbolic 3-manifolds.

The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Fefferman's invariant hypersurface measure.

We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes account of area, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces. The BV and measure properties of functions with finite relaxed energy are studied…

In this paper we describe a procedure for refining the given triangulation of a 3-manifold that scales the PL-metric according to a given weight function while creating no new normal surfaces. It is known that an incompressible surface $F$ in a triangulated 3-manifold $M$ is isotopic to a normal surface that is of mini…

Stable capillary surfaces in weighted balls are disks.

The paper proves smoothness of almost-minimizers' boundaries near the free boundary.

Explains the history and challenges of minimal surfaces.

For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…

Characterizes closures of mapping class group orbits on non-orientable surfaces.

Paper proves Harnack inequality for $f$-mean curvature flow.

We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space. Our main re…

Loewner inequality proven for curved surfaces.

Currents on cusped hyperbolic surfaces have a denseness property similar to compact surfaces.

We show that for any weakly convergent sequence of ergodic $SL_2(\mathbb{R})$-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech constants converge to the Siegel-Veech constant of the limit measure. Together with a measure equidistribution result due to Eskin-M…

Paper solves capillary Orlicz-Minkowski problem with new inequalities.

Stationary measures on hyperbolic surfaces with cusps are singular and stable under quasi-symmetries.

Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.

Let $X$ be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the covering group $π_1(X)$ on $\tilde{X}$ is of the first kind-i.e., the surface $X$ is equal to its convex core. We first prove that any geodesic lamination on $X$ is nowhere dense. Given a fixed geodesic pant…

Computes constants for cyclic covers of translation surfaces.

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

The paper proves that symmetric sets with minimal Gaussian surface area are nearly convex cylinders.

We prove the existence of a continuous $BV$ minimizer with $C^{0}$ boundary value for the $p$-area (pseudohermitian or horizontal area) in a parabolically convex bounded domain. We extend the domain of the area functional from $BV$ functions to vector-valued measures. Our main purpose is to study the first and second v…

We introduce a natural stratification of the space of projective classes of measured laminations on a complete hyperbolic surface of finite area. We prove a rigidity result, namely, the group of self-homeomorphisms of the space of projective measured laminations that preserve such a stratification is in general identif…

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

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…