arXiv research

Bound critical points for minimal Radó functions.

We construct the first and second Chern-Ricci functions on negatively curved minimal surfaces in ${\mathbb{R}}^{3}$ using Gauss curvature and angle functions, and establish that they become harmonic functions on the minimal surfaces. We prove that a minimal surface has constant first Chern-Ricci function if and only if…

Proves a function's locally least gradient property if its level sets are minimal laminations.

Winterbottom shape minimizes capillary functional under volume constraint.

Proof shows cones minimize certain geometric functionals.

Solves risk minimization problem with SSD constraints.

Study minimal timelike surfaces in 3D Lorentz-Minkowski space using holomorphic functions.

Submodular function minimization is well studied, and existing algorithms solve it exactly or up to arbitrary accuracy. However, in many applications, such as structured sparse learning or batch Bayesian optimization, the objective function is not exactly submodular, but close. In this case, no theoretical guarantees e…

The height function of various surfaces decomposes into finite sums of scaled and translated versions of itself.

Flat stable minimal hypersurfaces in 5 or 6D are always flat.

Paper classifies minimal graph transformations into new families of surfaces.

Research shows minimal communication limits adaptive function estimation rates.

Proves a principle for one-phase Bernoulli problem minimizers.

We consider the problem of minimizing the sum of submodular set functions assuming minimization oracles of each summand function. Most existing approaches reformulate the problem as the convex minimization of the sum of the corresponding Lovász extensions and the squared Euclidean norm, leading to algorithms requiring …

In this paper we construct complete simply connected minimal surfaces with a prescribed coordinate function. Moreover, we prove that these surfaces are dense in the space of all minimal surfaces with this coordinate function (with the topology of the smooth convergence on compact sets).

In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $\int \sqrt{1+K_γ^2} ds$, depending both on length and curvature $K$. We fix starting and ending points as well as initial and final directions. For this functional we discuss the problem o…

New algorithm for online convex minimization over integer lattice.

New functionals defined for free boundary minimal submanifolds in higher dimensions.

Study local minimizers of Ginzburg-Landau functionals in high dimensions, showing energy measures converge to rectifiable measures.

In this paper we raise the question whether every closed Riemannian manifold has a spine of minimal area, and we answer it affirmatively in the surface case. On constant curvature surfaces we introduce the spine systole, a continuous real function on moduli space that measures the minimal length of a spine in each surf…

The minimizer of a volume function is unique for klt singularities.

In this short note, we study the asymptotic property of Huisken's functional for mean curvature flow on the minimal submanifolds of Euclidean space. We prove that the limit of Huisken's functional equals to the extrinsic asymptotic volume ratio on the minimal submanifold of Euclidean space.

A space-like surface in Minkowski space-time is minimal if its mean curvature vector field is zero. Any minimal space-like surface of general type admits special isothermal parameters - canonical parameters. For any minimal surface of general type parameterized by canonical parameters we obtain Weierstrass representati…

Proves optimal regularity for sphere minimizers in 3-sphere.

Counts minimal tori in Riemannian manifolds with 6 or more dimensions.

Optimal Liouville theorem for minimal disks in any codimension.

The ultimate goal of optimization is to find the minimizer of a target function.However, typical criteria for active optimization often ignore the uncertainty about the minimizer. We propose a novel criterion for global optimization and an associated sequential active learning strategy using Gaussian processes.Our crit…

The conformal parameterisation of a minimal surface is harmonic. Therefore, a minimal surface is a critical point of both the energy functional and the area functional. In this paper, we compare the Morse index of a minimal surface as a critical point of the area functional with its Morse index as a critical point of t…

We prove that for any open Riemann surface $M$ and any non constant harmonic function $h:M \to \mathbb{R},$ there exists a complete conformal minimal immersion $X:M \to \mathbb{R}^3$ whose third coordinate function coincides with $h.$ As a consequence, complete minimal surfaces with arbitrary conformal structure and wh…

We consider adaptations of the Mumford-Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford-Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the…

We explore a connection between the Finslerian area functional and well-investigated Cartan functionals to prove new Bernstein theorems, uniqueness and removability results for Finsler-minimal graphs, as well as enclosure theorems and isoperimetric inequalities for minimal immersions in Finsler spaces. In addition, we …

The paper finds local minimizers for obstacle avoidance on curved spaces.

The paper shows how heat flow approximates area functional on specific geometric spaces.

First order methods can take extremely long to find global minima of non-convex functions.

The study proves that certain minimal surfaces are flat under specific conditions.

We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a dierence between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of ou…

We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods inherit the desirable convergence behavior of Newton-type methods for minimizing s…

Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total $(2k)$-th Gauss-Bonnet curvature function, called $(2k)$-minimal su…

In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most $(n-2),$ where $n$ is the dimension of its domain. Almgren used this result in an essential way to show t…

Since the pioneering work of Canham and Helfrich, variational formulations involving curvature-dependent functionals, like the classical Willmore functional, have proven useful for shape analysis of biomembranes. We address minimizers of the Canham-Helfrich functional defined over closed surfaces enclosing a fixed volu…

New learning algorithm for real analytic functions without gradient descent.

Study counts minimal Lagrangians in hyperbolic surfaces with precise growth rate.

This paper connects Laguerre minimal surfaces to Weierstrass representations.

We provide investment advice for an individual who wishes to minimize her lifetime poverty, with a penalty for bankruptcy or ruin. We measure poverty via a non-negative, non-increasing function of (running) wealth. Thus, the lower wealth falls and the longer wealth stays low, the greater the penalty. This paper general…

Study harmonic functions on submanifolds and their cones.

Optimal insurance minimizes ruin probability with non-decreasing functions.

The fact that minimal surfaces in the four-dimensional Euclidean space admit natural parameters implies that any minimal surface is determined uniquely up to a motion by two curvature functions, satisfying a system of two PDE's (the system of natural PDE's). In fact this solves the problem of Lund-Regge for minimal sur…

The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares $\sum_j(Y_j - μ(t_j))^2 + λ\int_a^b [μ"(t)]^2 dt$, where the data are $t_j,Y_j$, $j=1,..., n$. The minimization is taken over an infinite-dimensional function space, the space of all functions wi…