arXiv research

Sharp threshold found for Frechet mean of inhomogeneous graphs.

New graph properties inherited by Frechet mean and median.

We propose a novel framework for graph mean computation.

New method uses CNNs to estimate graph means.

Unified estimates for mean curvature in Lorentz-Minkowski space.

One of the most fundamental concepts in statistics is the concept of sample mean. Properties of the sample mean that are well-defined in Euclidean spaces become unwieldy or even unclear in graph spaces. Open problems related to the sample mean of graphs include: non-existence, non-uniqueness, statistical inconsistency,…

Survey on rigidity results for graphs with prescribed mean curvature.

We compute an approximate Fréchet mean for sets of sparse graphs.

Generalizes k-means to graphs using PageRank.

The paper computes an approximation to the sample Frechet mean of graph sets using spectral information.

Radial graphs with constant mean curvature found in Euclidean space.

Smoothness of graphs evolving by fractional mean curvature is proven.

Study of mean curvature flows on graphs in warped product manifolds, focusing on behavior at infinity.

Study on $t$-graphs with prescribed mean curvature in Heisenberg groups.

Interpolates mean shift and spectral clustering on graphs.

We extend the interior gradient estimate due to N. Korevaar and L. Simon for solutions of the mean curvature equation from the case of Euclidean graphs to the general case of Killing graphs. Our main application is the proof of existence of Killing graphs with prescribed mean curvature function for continuous boundary …

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

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 …

Proves existence of graph on torus with prescribed curvature.

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

Researchers prove constant mean curvature graphs in hyperbolic 3-space for specific domains.

We study a Neumann problem related to the evolution of graphs under mean curvature flow in Riemannian manifolds endowed with a Killing vector field. We prove that in a particular case these graphs converge to a bounded minimal graph which contacts the cylinder over the domain orthogonally along its boundary.

Proves existence and uniqueness of Killing graphs with prescribed curvature.

In this note we study a large class of mean curvature type flows of graphs in product manifold $N\times R$ where N is a closed Riemann- ian manifold. Their speeds are the mean curvature of graphs plus a prescribed function. We establish long time existence and uniformly convergence of those flows with a barrier conditi…

It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian manifolds.

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

Symmetric graphs flow without singularities on their axis.

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…

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…

This work represents an application of constant mean curvature graphs (as solutions of the mean curvature PDE) to non-linear non-Darcy flows in porous media. It relates time invariant pressure distribution graphs to graphs of constant mean curvature surfaces. This differential geometric interpretation provides an impor…

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…

The paper proves uniqueness of evolving graphs by mean curvature flow under specific conditions.

In this paper, we give a height estimate for constant mean curvature graphs. Using this result we prove two results of uniqueness for the Dirichlet problem associated to the constant mean curvature equation on unbounded domains.

We prove an existence result for helicoidal graphs with prescribed mean curvature in a large class of warped product spaces which comprises space forms.

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

New method discovers mean and variance causal graphs from heteroscedastic data.

We study the task of semi-supervised learning on multilayer graphs by taking into account both labeled and unlabeled observations together with the information encoded by each individual graph layer. We propose a regularizer based on the generalized matrix mean, which is a one-parameter family of matrix means that incl…

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.

A new metric compares true and learned causal graphs considering data and graph structure.

The study shows that certain graphs are regular at boundary points.

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.

It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field.

We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in three-dimensional Lorentzian Killing submersions. Our duality extends the Calabi correspondence between minimal graphs in the Euclid…

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 prove the existence and uniqueness of graphs with prescribed mean curvature function in a large class of Riemannian manifolds which comprises spaces endowed with a conformal Killing vector field.

Multilayer graphs encode different kind of interactions between the same set of entities. When one wants to cluster such a multilayer graph, the natural question arises how one should merge the information different layers. We introduce in this paper a one-parameter family of matrix power means for merging the Laplacia…

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…

Study on mean curvature flow of graphs in higher dimensions.