arXiv research

Sharp area estimates for minimal submanifolds in curved spaces.

The paper estimates area and volume for spacetimes with integral mean curvature bounds.

Random forests and LASSO methods improve small area estimation using auxiliary data.

Sharp estimate for flow in any dimension.

Study minimal annuli in a slab, estimating their area.

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

Horseshoe priors improve small area estimation by borrowing strength globally but locally.

Estimate sphere area in Sol group up to a factor of 10.

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…

Paper estimates area covered by a line-sweep sensor in robotics.

Study estimates hypersurface areas in curved spaces, with applications to spectrum bounds.

We estimate whether there is an embedding from one n-dimensional rectangle into another which expands every k-dimensional area. Our estimate is sharp up to a constant factor in each dimension.

Study proves area estimates for stable capillary hypersurfaces with nonpositive Yamabe invariant.

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…

Minimal surfaces' area bounds proven equivalent, extending known results.

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

JAPAN uses flow-based models to create adaptive prediction areas with better coverage guarantees.

In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area minimizing currents in a Riemannian manifold, with a slight improvement on the regularity assumption for the latter. This note establishes a new a priori estimate on the excess measure of an a…

We show an Uhlenbeck type estimate for closed simply connected manifolds which provides the existence of certain exact sequences in K-area homology. This leads to the behavior of the K-area homology under surgery. Moreover, we give an index theoretic obstruction to positive scalar curvature on compact spin manifolds wi…

In this paper we consider the evolution of a graph-like hypersurface by anisotropic mean curvature flow, under some restrictions on the anisotropic area integrand. We find interior estimates (in both time and space) on the gradient of such hypersurfaces, depending only on the height of the graph and the anisotropic are…

In this paper we consider surfaces which are critical points of the Willmore functional subject to constrained area. In the case of small area we calculate the corrections to the intrinsic geometry induced by the ambient curvature. These estimates together with the choice of an adapted geometric center of mass lead to …

The paper proves optimal systolic inequalities for Möbius strip and Klein bottle.

We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In particular, we prove that if we are given a sequence of closed minimal hypersurfaces of …

The paper studies $\mathfrak{g}$-stability of surfaces with boundary and derives area estimates.

PriorVAE uses VAEs to efficiently encode spatial priors for small-area estimation.

Starting at a saddle tower surface, we give a new existence proof of the Lawson surfaces $ξ_{m,k}$ of high genus by deforming the corresponding DPW potential. As a byproduct, we obtain for fixed $m$ estimates on the area of $ξ_{m,k}$ in terms of their genus $g=m k \gg1$.

In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a 3-manifold under the Ricci flow. This estimate generalizes an area estimate of Hamilton, which we will recall in the first part of t…

The article disproves a local systolic inequality and shows a lower bound on filling area.

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 …

The paper studies volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.

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

For a given boundary set consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface spanning the boundary.

In this article, we investigate the geometry of critical metrics of the volume functional on an $n$-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifol…

This paper considers some fundamental questions concerning marginally trapped surfaces, or apparent horizons, in Cauchy data sets for the Einstein equation. An area estimate for outermost marginally trapped surfaces is proved. The proof makes use of an existence result for marginal surfaces, in the presence of barriers…

New estimate for Curve Shortening Flow improves graphical solutions.

We analyze the level sets of the norm of the Witten spinor in an asymptotically flat Riemannian spin manifold of positive scalar curvature. Level sets of small area are constructed. We prove curvature estimates which quantify that, if the total mass becomes small, the manifold becomes flat with the exception of a set o…

This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to expl…

We formalize AURC and develop estimators for SC systems.

In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for multi-dimensional SDEs driven by Brownian motions. Giles has previously shown that if we combine a numerical approximation with strong order of convergence $O(Δt)$ with MLMC we can reduce the computational complexity to estimate expected value…

We propose a novel classifier accuracy metric: the Bayesian Area Under the Receiver Operating Characteristic Curve (CBAUC). The method estimates the area under the ROC curve and is related to the recently proposed Bayesian Error Estimator. The metric can assess the quality of a classifier using only the training datase…

We give an explicit estimate of the area of a closed surface by the diameter and a lower bound of curvature. This is better than Calabi-Cao's estimate for a nonnegatively curved two-sphere.

In this paper, we consider the area-preserving mean curvature flow with free Neumann boundaries. We show that for a rotationally symmetric $n$-dimensional hypersurface in $\R^{n+1}$ between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the m…

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…

New algorithm speeds up online mapping of unknown terrains.

The paper studies minimal surfaces in 3D spheres and balls, confirming conjectures and identifying new surfaces.

Recent reinforcement learning algorithms, though achieving impressive results in various fields, suffer from brittle training effects such as regression in results and high sensitivity to initialization and parameters. We claim that some of the brittleness stems from variance differences, i.e. when different environmen…

Kriformer uses graph transformers to estimate data in sparse sensor areas.

New translations defined; curve shortening flow solved in hyperbolic plane.