arXiv research

Improved understanding of translating solitons using new techniques.

Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.

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

We summarize results concerning the Bernstein property of differential equations.

In this paper, we prove some Bernstein type results for $n$-dimensional minimal Lagrangian graphs in quaternion Euclidean space $H^n\cong R^{4n}$. In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in $C^n$

Study Bernstein results for self-shrinking solutions in Lagrangian flow.

We establish Bernstein Theorems for Lagrangian graphs which are Hamiltonian minimal or have conformal Maslov form. Some known results of minimal (Lagrangian) submanifolds are generalized.

The paper solves Bernstein problems for specific submanifolds in high-dimensional spaces.

Adaptive Bernstein copulas improve risk management by preventing overfitting and reducing simulation effort.

In this paper we study nonconvex penalization using Bernstein functions. Since the Bernstein function is concave and nonsmooth at the origin, it can induce a class of nonconvex functions for high-dimensional sparse estimation problems. We derive a threshold function based on the Bernstein penalty and give its mathemati…

Proves properties of hypersurfaces in Euclidean space.

In this paper we study nonconvex penalization using Bernstein functions whose first-order derivatives are completely monotone. The Bernstein function can induce a class of nonconvex penalty functions for high-dimensional sparse estimation problems. We derive a thresholding function based on the Bernstein penalty and di…

Improves Bernstein theorem for zero mean curvature hypersurfaces in Lorentz-Minkowski space.

We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.

Ancient solutions to mean curvature flow have unique shapes.

This is a survey of our work on spacelike graphic submanifolds in pseudo-Riemannian products, namely on Heinz-Chern and Bernstein-Calabi results and on the mean curvature flow, with applications to the homotopy of maps between Riemannian manifolds.

The study models insurance dependence using Bernstein copulas.

New theorems prove uniqueness of solutions to geometric PDEs.

Explains Bernstein theorems for various geometric PDEs.

Improved analysis of UCRL2 with empirical Bernstein inequality reduces exploration-exploitation regret.

Ancient symplectic solutions to mean curvature flow are flat.

We present Bernstein-Sato identities for scalar-, spinor- and differential form-valued distribution kernels on Euclidean space associated to conformal symmetry breaking operators. The associated Bernstein-Sato operators lead to partially new formulae for conformal symmetry breaking differential operators on functions, …

Minimal surface equation results in constant solutions on RCD spaces.

Study on uniqueness of hypersurfaces in hyperbolic space with constant mean curvature.

Article provides Bernstein gradient estimates for heat equations with potential terms.

Bernstein theorem proven for 2-valued minimal graphs in 4D.

A celebrated result of S. Bernstein states that every solution of the minimal surface equation over the entire plane has to be an affine linear function. Since the paper of Bernstein appeared in 1927, many different proofs and generalizations of this beautiful theorem were given, namely to higher dimensions and to more…

Self-shrinkers are important geometric objects in the study of mean curvature flows, while the Bernstein Theorem is one of the most profound results in minimal surface theory. We prove a Bernstein type result for graphical self-shrinker surfaces with codimension two in $\mathbb{R}^4$. Namely, under certain natural cond…

The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.

Improves Bernstein theorem for space-like graphs in Lorentz-Minkowski space.

Sharp inequalities for matrix means with unknown variance.

For entire spacelike stationary 2-dimensional graphs in Minkowski spaces, we establish Bernstein type theorems under specific boundedness assumptions either on the W-function or on the total (Gaussian) curvature. These conclusions imply the classical Bernstein theorem for minimal surfaces in 3-dimensional Euclidean spa…

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hy…

In this paper, we study the properties of potential function of the translating soliton $M$ in $R^{n+1}$ and the volume growth of the intersection of Euclidean balls with $M$. We give a condition to obtain the Bernstein theorem for the translating solitons. We also give an outline of a simple proof of the Bernstein the…

Paper proves stable minimal surfaces in 3D are flat.

We obtain a Bernstein theorem for special Lagrangian graphs in n-dimensional complex space for arbitrary n only assuming bounded slope, but no quantitative restriction.

Minimal hypertori found in 4D sphere, solving Bernstein conjecture.

We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that expanders with vanishing relative entropy are unique in a generic sense. This also impl…

New inequality for ternary variables improves on existing measures.

In this paper we provide several uniqueness and non-existence results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian warped products under appropriate geometric assumptions. As a consequence of this parametric study, we obtain very general uniqueness and non-existence results for a…

New maximal surfaces solve Bernstein problems.

New PAC-Bayesian bound improves generalization performance.

Based on a calibration argument, we prove a Bernstein type theorem for entire minimal graphs over Gauss space $\mathbb{G}^n$ by a simple proof.

In this note, we propose Bernstein's problem and De Giorgi's conjecture for spatially inhomogeneous equations, as well as De Giorgi's conjecture for system of reaction-diffusion equations.

The study proves planes are the only complete uniformly elliptic Weingarten multigraphs.

New minimal hypersphere found in 4-sphere solving Bernstein problem.

The study proves that certain constant mean curvature surfaces in a specific cone are either spheres or horospheres.

Study introduces new Bernstein inequalities for dependent data in Hilbert spaces.