arXiv research

Paper proves stability of positive mass theorem for specific types of manifolds.

The study examines the graphical mean curvature flow on compact manifolds with bounded bi-Ricci curvature.

Develops a graphical calculus for stable curvature invariants.

Graphically discrete groups have strong rigidity properties.

We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and deri…

Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical convergence away from finitely many points. We show that the limit of a sequence …

Introduce new boundary mass for asymptotically flat half-manifolds

The long-time existence and umbilicity estimates for compact, graphical solutions to expanding curvature flows are deduced in Riemannian warped products of a real interval with a compact fibre. Notably we do not assume the ambient manifold to be rotationally symmetric, nor the radial curvature to converge, nor a lower …

We present a quasi-local version of the stability of the positive mass theorem. We work with the Brown--York quasi-local mass as it possesses positivity and rigidity properties, and therefore the stability of this rigidity statement can be studied. Specifically, we ask if the Brown--York mass of the boundary of some co…

Study preserves planar and graphical properties of curves under elastic flow.

The paper proves rigidity results for manifolds with nonnegative scalar curvature.

The rigidity of the Positive Mass Theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We study the stability of this statement for spaces that can be realized as graphical hypersurfaces in Euclidean space. We prove (under certain technical…

In this article we prove two non-existence results for translating solitons of the mean curvature flow (translators for short) in $\mathbb{R}^{m+1}$. We also obtain an upper bound to the maximum height that a compact embedded translator in $\mathbb{R}^{3}$ can achieve. On the other hand, we study graphical perturbation…

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any such compact minimal surface $M$, we provide a priori radius estimate which depend…

Ancient curve flows classified into specific types.

In this paper, we can prove the existence and uniqueness of solutions to the constant mean curvature (CMC for short) equation with nonzero Neumann boundary data in product manifold $M^{n}\times\mathbb{R}$, where $M^{n}$ is an $n$-dimensional ($n\geq2$) complete Riemannian manifold with nonnegative Ricci curvature, and …

In this paper, we prove a positive mass theorem and Penrose-type inequality of the Gauss-Bonnet-Chern mass $m_2$ for the graphic manifold with flat normal bundle.

New method learns dependencies in high-dimensional data without graph assumptions.

AGMs outperform EGMs in generalizing to unseen inference tasks.

The covariance structure of multivariate functional data can be highly complex, especially if the multivariate dimension is large, making extensions of statistical methods for standard multivariate data to the functional data setting challenging. For example, Gaussian graphical models have recently been extended to the…

We give an explicit formula for the Gauss-Bonnet-Chern mass of an asymptotically flat graphical manifold of arbitrary codimension and use it to prove the positive mass theorem and the Penrose inequality for graphs with flat normal bundle.

We give, via elementary methods, explicit formulas for the ADM mass which allow us to conclude the positive mass theorem and Penrose inequality for a class of graphical manifolds which includes, for instance, that ones with flat normal bundle.

We study minimal graphic functions on complete Riemannian manifolds $\Si$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with …

Compactness proven for CMC surfaces with bounded topology and boundary length.

Stability of positive mass theorem for hyperbolic manifolds studied.

AdaRL adapts quickly to new environments with minimal data.

Study of mean curvature flow in warped products preserving equivariance.

We analyze variational inference for highly symmetric graphical models such as those arising from first-order probabilistic models. We first show that for these graphical models, the tree-reweighted variational objective lends itself to a compact lifted formulation which can be solved much more efficiently than the sta…

This paper is concerned with data-driven unsupervised domain adaptation, where it is unknown in advance how the joint distribution changes across domains, i.e., what factors or modules of the data distribution remain invariant or change across domains. To develop an automated way of domain adaptation with multiple sour…

The paper defines capacities for minimal graphs over manifolds and proves the half-space property.

Let f:Σ_1 --> Σ_2 be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in the product of Σ_1 and Σ_2 by the mean curvature flow. Under suitable conditions on the curvature of Σ_1 and Σ_2 and the differential of the initial map, we show that the flow…

We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold $M$ with only one end if $M$ has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constan…

In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric quantities that yield preservation of certain qu…

Bayesian method learns network structure from Gaussian process priors.

Develops graphical calculus for monoidal categories with twisted pivotal structures.

Develops an efficient method for real-time data analysis and visualization.

QT improves inference in complex PGMs with hidden variables.

Let $Σ$ be a complete Riemannian manifold with the volume doubling property and the uniform Neumann-Poincar$\mathrm{\acute{e}}$ inequality. We show that any positive minimal graphic function on $Σ$ is a constant.

Graphical lasso may fail to fit models when data points are insufficient.

Study on rigidity of translating hypersurfaces not in graphical direction.

This is a short description of graphic lambda calculus, with special emphasis on a duality suggested by the two different appearances of knot diagrams, in lambda calculus and emergent algebra sectors of the graphic lambda calculus respectively. This duality leads to the introduction of the dual of the graphic beta move…

Extends a theorem for first-order elliptic operators on manifolds.

Self-shrinkers are hypersurfaces that shrink homothetically under mean curvature flow; these solitons model the singularities of the flow. It it presently known that an entire self-shrinking graph must be a hyperplane. In this paper we show that the hyperplane is rigid in an even stronger sense, namely: For $2 \leq n \…

Paper proves unique tangent flow at infinity for entropy-limited curve shortening.

We show how the theory of tangles is equivalent to that of well-connected tangles. These are drawn on a surface with boundary, and equivalent via Reidemeister moves of a restricted kind. This reworking of the graphical foundations for link and tangle theory can be expected to have a variety of applications, including o…

We consider the problem of learning high-dimensional Gaussian graphical models. The graphical lasso is one of the most popular methods for estimating Gaussian graphical models. However, it does not achieve the oracle rate of convergence. In this paper, we propose the graphical nonconvex optimization for optimal estimat…

Quantum graphical models (QGMs) extend the classical framework for reasoning about uncertainty by incorporating the quantum mechanical view of probability. Prior work on QGMs has focused on hidden quantum Markov models (HQMMs), which can be formulated using quantum analogues of the sum rule and Bayes rule used in class…

New method for tuning Graphical Lasso hyperparameters.