arXiv research

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…

Overview of affine surface area and its history.

The paper connects least area surfaces to quasi-normal surfaces in 3-manifolds.

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for $L_φ$ affine surface areas are established.

Introduces new weighted floating functions and affine surface areas.

Upper bounds on area for surfaces with constant mean curvature in hyperbolic 3-manifolds.

We study an area minimization problem for spacelike zero mean curvature surfaces in four dimensional Lorentz-Minkowski space. The areas of these surfaces are compared of with the areas of certain marginally trapped surfaces having the same boundary values.

Study on existence and structure of P-area surfaces in Heisenberg group.

Minimal surfaces in hyperbolic space have a renormalized area criterion.

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

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

Minimal surfaces in hyperbolic space have a sharp area bound.

We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.

Minimal surfaces in a ball have limited area.

We study the classification of area-stationary and stable $C^2$ regular surfaces in the space of the rigid motions of the Minkowski plane E(1,1), equipped with its sub-Riemannian structure. We construct examples of area-stationary surfaces that are not foliated by sub-Riemannian geodesics. We also prove that there exis…

Generalized Cauchy's surface area formula to arbitrary submanifolds in R^n.

We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carathéodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero (or constant if a volume-preservin…

New minimal surfaces grow area very quickly.

In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $p\neq -n$. Our definitions are motivated from an equivalent formula for the mixed $p$-affine surface area. Some properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are prove…

The paper develops inequalities for log-concave functions and related surface areas.

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

Geodesics on hyperbolic surfaces become evenly spread over time.

We study surfaces in TN that are area-stationary with respect to a neutral Kaehler metric constructed on TN from a riemannian metric g on N. We show that holomorphic curves in TN are area-stationary, while lagrangian surfaces that are area-stationary are also holomorphic and hence totally null. However, in general, are…

We obtain a bound for the area of a capillary $H-$surface in a three-manifold with umbilic boundary and controlled sectional curvature. We then analyze the geometry when this area bound is realized, and obtain rigidity theorems. As a side product, we obtain existence of totally geodesic embedded surfaces in hyperbolic …

New bounds on genus and area for CMC surfaces in 3-manifolds.

We show that area minimizing polyhedral surfaces are saddle.

Solves area-minimizing surface problem for finite curves in H^2xR.

In this paper we study areas (called p-areas) and volumes for parametric surfaces in the 3D-Heisenberg group $\mathbb{H}_1$, which is considered as a flat model of pseudo-hermitian manifolds. We derive the formulas of p-areas and volumes for parametric surfaces in $\mathbb{H}_1$ and show that the classical result of Pa…

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

Study of large area-constrained Willmore surfaces in Schwarzschild-like manifolds.

Characterizes area-minimizing maps for surfaces of genus ≥ 2.

Proves unique continuation for area minimizing currents.

Improved bound on the product of first Laplacian eigenvalue and area for genus three surfaces.

Formula for Heisenberg group surface areas derived.

The paper finds metrics for surfaces with boundaries that match specific eigenvalues and areas.

The paper proves a diastolic inequality linking surface area and loop length.

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

Study on ratio of intrinsic to extrinsic metrics and its relation to surface area.

Short proof shows infinite diameter for surface diffeomorphisms.

We prove prove a bridge principle at infinity for area-minimizing surfaces in the hyperbolic space $\mathbb{H}^3$, and we use it to prove that any open, connected, orientable surface can be properly embedded in $\mathbb{H}^3$ as an area-minimizing surface. Moreover, the embedding can be constructed in such a way that t…

Researchers study surface area functionals in CR manifolds, deducing equations for various cases.

For appropriately values of $H$, we obtain an area estimate for a complete non-compact $H$-surface of finite topology and finite area, embedded in a three-manifold of negative curvature. Moreover, in the case of equality and under additional assumptions, we prove that a neighbourhood of the mean convex side of the surf…

Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above by the area and the topology of the surface. Combining these results with work of…

Motivated by classical theorems on minimal surface theory in compact hyperbolic three-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic three-manifold of finite volume. We prove any closed immersed incompressible surface can be deform…

A well known consequence of the Wirtinger inequality is that in a Kaehler surface a holomorphic curve is an area minimizer in its homology class. In light of this result it is natural, given a Kaehler surface, to investigate the relation between area minimizers and complex curves. When the Kaehler surface is a K3 surfa…

We define the ``volume'' contained by pointed $k$-surfaces, first studied by the author in [9], and we show that this volume is always finite. Likewise, we show that the surface area of a pointed $k$-surface is always finite.

Motivated by the HRRT-formula for holographic entanglement entropy, we consider the following question: what are the position and the surface area of extremal surfaces in a perturbed geometry, given their anchor on the asymptotic boundary? We derive explicit expressions for the change in position and surface area, ther…

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