arXiv research

Develops a method to construct entire minimal graphs of odd dimensions.

It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ is called of mixed type if it changes causal type from space-like to time-like. In $\boldsymbol{R}^3_1$, Osamu Kobayashi found …

Classifies solitons for surface diffusion flow of graphs.

New method embeds entire graphs without supervision.

New examples of mixed-type zero-curvature graphs found.

Entire minimal graphs in Heisenberg space have negative Gauss curvature.

The paper proves rigidity for self-shrinking spacelike graphs in pseudo-Euclidean space.

Study inverse mean curvature flow on entire graphs, proving finite time existence for certain asymptotic cases.

We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil_3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil_3 is entire. Also, the si…

In this paper, we provided conditions for an entire constant mean curvature Killing graph lying inside a possible unbounded region to be necessarily a slice.

We study convex entire graphs evolving with normal velocity equal to a positive power of the mean curvature. Under mild assumptions we prove longtime existence.

We prove that a family of entire intrinsic minimal graphs in the Heisenberg group are not perimeter minimizing.

We show that under certain curvature conditions of the ambient space an entire Killing graph of constant mean curvature lying inside a slab must be a totally geodesic slice.

We obtain an optimal estimate for the extrinsic curvature of an entire minimal graph in $\H^2\times\R$, $\H^2$ the hyperbolic plane.

In this note we prove that every two-dimensional entire Willmore graph in $R^3$ with square integrable mean curvature is a plane.

Proposes an unsupervised graph neural network for entire graph representation.

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.

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is sm…

We show that the number of entire maximal graphs with finitely many singular points that are conformally equivalent is a universal constant that depends only on the number of singularities, namely 2^$ for graphs with n+1 singularities. We also give an explicit description of the family of entire maximal graphs with a f…

In the lorentzian product $\Bbb G^n\times\Bbb R_1,$ we give a comparison between the $f$-volume of an entire $f$-maximal graph and the $f$-volume of the hyperbolic $H_r^+$ under the assumption that the gradient of the function defining the graph is bounded away from 1. As a consequence, we obtain a Bernstein type theor…

We present two initial graphs over the entire $\mathbb{R}^n$, $n \geq 2$ for which the mean curvature flow behaves differently from the heat flow. In the first example, the two flows stabilize at different heights. With our second example, the mean curvature flow oscillates indefinitely while the heat flow stabilizes. …

The study proves properties of capillary graphs in half-spaces.

In this note, we prove that smooth self-shrinkers in $\Real^{n+1}$, that are entire graphs, are hyperplanes. Previously Ecker and Huisken showed that smooth self-shrinkers, that are entire graphs and have at most polynomial growth, are hyperplanes. The point of this note is that no growth assumption at infinity is need…

The study constructs non-invariant H-graphs in hyperbolic space with specific asymptotic boundaries.

Unlike $\mathbb{R}^{3}$, the homogeneous spaces $\mathbb{E}(-1,τ)$ have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantees that a minimal surface in $\mathbb{E}(-1,τ)$ is such a graph. More specifically: we introduce the definition of a generalized slab in $\mathbb{E…

In this paper, we prove a half-space theorem with respect to constant mean curvature $1/2$ entire graphs in $\mathbb{E(-1,τ)}$. If $Σ$ is such an entire graph and $Σ'$ is a properly immersed constant mean curvature $1/2$ surface included in the mean convex side of $Σ$ then $Σ'$ is a vertical translate of $Σ$. We also h…

We show that a properly immersed minimal hypersurface in M x R_+ equals some M x {c} when M is a complete, recurrent n-dimensional Riemannian manifold with bounded curvature. If on the other hand, M has nonnegative Ricci curvature with curvature bounded below, the same result holds for any positive entire minimal graph…

In this paper, we obtain an Ecker-Huisken type result for entire graphs with parallel mean curvature.

Minimal graphs grow slowly on curved spaces, proving constant solutions.

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

Paper finds space-like maximal surfaces with entire null lines in 3D space-time.

The paper solves a curvature equation for a specific spacetime foliation.

We obtain a Bernstein type result for entire two dimensional minimal graphs in $\mathbb{R}^{4}$, which extends a previous one due to L. Ni. Moreover, we provide a characterization for complex analytic curves.

Constructs graphs with singularities in a special space.

Develops a duality for graphs in Riemannian and Lorentzian spaces with prescribed mean curvature.

Inductive graph-based approach for disease classification with incomplete data.

Let S be a C^2 H-minimal noncharacteristic hypersurface in the first Heisenberg group. We show that if S contains a graphical strip, then it is not a stable minimal surface. Moreover, we show that if S is a C^2 H-minimal noncharacteristic entire graph which is not itself a vertical plane, then S contains a graphical st…

Shapley Flow interprets model predictions using a graph-based approach to feature importance.

Unified estimates for mean curvature in Lorentz-Minkowski space.

We find a set of generators for the automorphism group of a graph product of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup of star-automorphisms defined in [7]. We follow closely the plan of M. Laurence's paper [11].

The aim of this work is studying translating graphs by mean curvature flow in $\Real^3$. We prove non-existence of complete translating graphs over bounded domains in $\Real^2$. Furthermore, we show that there are only three types of complete translating graphs in $\Real^3$; entire graphs, graphs between two vertical p…

Integrates learning and optimization on graphs, improving prediction accuracy.

Study shows stable graphs in Heisenberg group are essentially planes.

We prove that the ends of a properly immersed simply or one connected minimal surface in H(2)xR contained in a slab of height less than πof H(2)xR, are multi-graphs. When such a surface is embedded then the ends are graphs. When embedded and simply connected, it is an entire graph.

Existence and uniqueness in ${\Bbb R}^{n,1}$ of entire spacelike hypersurfaces contained in the future of the origin $O$ and asymptotic to the light-cone, with scalar curvature prescribed at their generic point $M$ as a negative function of the unit vector $\overrightarrow{Om}$ pointing in the direction of $\overrighta…

GraphTEE estimates treatment effects on graph-structured targets, mitigating bias.

Proposes clique pooling for graph classification.

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