arXiv research

The study examines the graphical mean curvature flow on compact manifolds with bounded bi-Ricci curvature.

We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly increase the first eigenvalue normalized by area if the initial su…

Sharp estimate for flow in any dimension.

Topological entropy decreases strictly along Ricci flow near hyperbolic metrics.

We develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a "long neck principle" for a compact Riemannian spin $n$-manifold with boundary $X$, stating tha…

We discuss a special class of solutions to the minimal surface system. These are vector-valued functions that "decrease area" and are natural generalization of scalar functions. After defining area-decreasing maps, we show several classical results for the minimal surface equation can be generalized. We also conjecture…

We consider the mean curvature flow of the graph of a smooth map $f:\mathbb{R}^2\to\mathbb{R}^2$ between two-dimensional Euclidean spaces. If $f$ satisfies an area-decreasing property, the solution exists for all times and the evolving submanifold stays the graph of an area-decreasing map $f_t$. Further, we prove unifo…

In this note, we show that the solution to the Dirichlet problem for the minimal surface system in any codimension is unique in the space of distance-decreasing maps. This follows as a corollary of the following stability theorem: if a minimal submanifold $Σ$ is the graph of a (strictly) distance-decreasing map, then $…

Let $M$ be a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function. We show that the isoperimetric profile of $M$ is a continuous and non-decreasing function. Particular cases are Hadamard manifolds and complete non-compact manifolds with strictly positive sectional curvatur…

We decrease the $rms$ mean curvature and area of a variable surface with a fixed boundary by iterating a few times through a curvature-based variational algorithm. For a boundary with a known minimal surface, starting with a deliberately chosen non-minimal surface, we achieve up to 65 percent of the total possible decr…

Let $f:M\to N$ be a smooth area decreasing map between two Riemannian manifolds $(M,\gm)$ and $(N,\gn)$. Under weak and natural assumptions on the curvatures of $(M,\gm)$ and $(N,\gn)$, we prove that the mean curvature flow provides a smooth homotopy of $f$ to a constant map.

Analyzed geometric and diffusion properties of a coupled system.

Maps on foliated manifolds decrease area and scalar curvature is negative.

In this article we give a complete description of the evolution of an area decreasing map $f:M\to N$ induced by its mean curvature in the situation where $M$ and $N$ are complete Riemann surfaces with bounded geometry, $M$ being compact, for which their sectional curvatures $σ_M$, $σ_N$ satisfy $\minσ_M\ge\supσ_N$.

New proof of Llarull's rigidity theorem in odd dimensions via spectral flow.

The study proves that certain stable minimal hypersurfaces must be cylindrical.

Study on mean curvature flow of graphs in higher dimensions.

In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in the boundary of M. Our main result is that for any g>=0…

We prove that a strictly stable constant-mean-curvature hypersurface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L^1 neighborhood.

In this work we classify the stable regions (second order minima of perimeter under an area constraint) in tori of revolution with piecewise continuous decreasing Gauss curvature from the longest parallel and with a horizontal symmetry. Some applications to isoperimetric problems are also given.

It is well known that the area $U$ of the triangle formed by three tangents to a parabola $X$ is half of the area $T$ of the triangle formed by joining their points of contact. In this article, we study some properties of $U$ and $T$ for strictly convex plane curves. As a result, we establish a characterization for par…

We prove that a strictly stable minimal $C^2_h$ intrinsic graph G is locally area-minimizing, i.e. given any $C^1_h$ graph $S$ with the same boundary, $\text{Area}(G)<\text{Area}(S)$ unless $G=S$. As a consequence we show the existence and the uniqueness of $C^\infty$ minimal graphs with prescribed small boundary datum…

Study examines Hilbert area of inscribed polygons in projective geometry.

Let $\big(M,g^{TM}\big)$ be a noncompact complete spin Riemannian manifold of even dimension $n$, with $k^{TM}$ denote the associated scalar curvature. Let $f\colon M\rightarrow S^{n}(1)$ be a smooth area decreasing map, which is locally constant near infinity and of nonzero degree. We show that if $k^{TM}\geq n(n-1)$ …

New curve flow preserves area and converges to a circle.

We study the problem of finding a minimal graph with prescribed boundary data in arbitrary dimension and codimension. Existence, uniqueness, stability and regularity are treated. We first present the well-known results for codimension one: Jenkins-Serrin's existence theorem, convexity properties of the area which give …

We propose a global invariant $σ_c$ for contact manifolds which admit a strictly pseudoconvex CR structure, analogous to the Yamabe invariant $σ$. We prove that this invariant is non-decreasing under handle attaching and under connected sum. We then give a lower bound on $σ_c$ in a particular case.

Study characterizes hulls and capacities on Riemannian manifolds, proving isoperimetric inequalities.

Let N be a complete, simply-connected surface of constant curvature κ\leq 0. Moreover, suppose that Ωand \tildeΩ are strictly convex domains in N with the same area. We show that there exists an area-preserving diffeomorphism from Ωto \tildeΩ whose graph is a minimal submanifold of N \times N.

This note shows surfaces in stable 3D data are bounded by area and diameter.

It is well known that the area $U$ of the triangle formed by three tangents to a parabola $X$ is half of the area $T$ of the triangle formed by joining their points of contact. In this article, we consider whether this property and similar ones characterizes parabolas. As a result, we present three conditions which are…

We prove that every stationary polyhedral varifold minimizes area in the following senses: (1) its area cannot be decreased by a one-to-one Lipschitz ambient deformation that coincides with the identity outside of a compact set, and (2) it is the varifold associated to a mass-minimizing flat chain with coefficients in …

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…

New SAGA algorithm with decreasing step for stochastic optimization.

Archimedes knew that the area between a parabola and any chord $AB$ on the parabola is four thirds of the area of triangle $ΔABP$ where P is the point on the parabola at which the tangent is parallel to $AB$. We consider whether this property (and similar ones) characterizes parabolas. We present five conditions which …

4-manifolds with nonnegative sectional curvature are area-extremal.

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…

There is a small number of case studies of automatic land cover classification on the coastal area. Here, I test extraction of seagrass beds, sandy area, oyster farming rafts at Mangoku-ura Lagoon, Miyagi, Japan by comparing manual tracing, simple image segmentation, and image transformation using deep learning. The re…

We prove some related results concerning blow-up solutions for the Jang equation. First: it has been shown that, given an outermost marginally outer trapped surface (MOTS) Σ, there exists a solution to Jang's equation which blows up at Σ. Here we show that in addition, large classes of spherically symmetric initial dat…

Paper constructs flows converging to cones and foliations.

BCD algorithm finds global minima in neural networks.

This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic functi…

Method bounds tail probabilities of continuous RVs.

Archimedes showed that the area between a parabola and any chord $AB$ on the parabola is four thirds of the area of triangle $ΔABP$, where P is the point on the parabola at which the tangent is parallel to the chord $AB$. Recently, this property of parabolas was proved to be a characteristic property of parabolas. With…

In this paper we study the curvature flow of a curve in a plane endowed with a minkowskian norm whose unit ball is smooth. We show that many of the properties known in the euclidean case can be extended (with due adaptations) to this new situation. In particular, we show that simple, closed, strictly convex, smooth cur…

Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A corollary is that any area-decreasing map between unit spheres (of possibly different dimensions) is homotopic to a constant…

Proves rigidity of maps between manifolds with scalar curvature constraints.

Weyl energy decreases for connected sums of certain four-manifolds.