arXiv research

Study mean curvature flow in hyperbolic 3-manifolds, proving foliations and existence of minimal surfaces.

The paper proves unique geodesics on hyperbolic surfaces and finds lower bounds.

Let $S$ be a compact hyperbolic Riemann surface of genus $g \geq 2$. We call a systole a shortest simple closed geodesic in $S$ and denote by $\mathop{sys}(S)$ its length. Let $\mathop{msys(g)}$ be the maximal value that $\mathop{sys}(\cdot)$ can attain among the compact Riemann surfaces of genus $g$. We call a (global…

Study confirms a 2-sphere metric with three geodesics of minimal length.

Estimates the degree of trace fields of hyperbolic Dehn fillings.

Study on spectral gaps of hyperbolic surfaces as genus increases.

New theorem finds new minimal hypersurfaces in hyperbolic space.

Max-pooling architectures are theoretically analyzed and shown to be globally optimized and generalize well.

In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. We show that there exists a class of initial velocities such that the solution of the corresponding initial value problem exists only at a finite time interval $[0,T_{\max})$ and when $t$ goes to $T_{\max}$, the sol…

We study the implicit bias of gradient descent methods in solving a binary classification problem over a linearly separable dataset. The classifier is described by a nonlinear ReLU model and the objective function adopts the exponential loss function. We first characterize the landscape of the loss function and show th…

New findings on the max margin problem in neural networks.

Affine deformations of convex cones yield special spacetime structures.

New groups found with critical exponents close to but less than max.

Generalizes global hyperbolicity to higher signatures and proves compactness.

In this paper, we investigate two hyperbolic flows obtained by adding forcing terms in direction of the position vector to the hyperbolic mean curvature flows in \cite{klw,hdl}. For the first hyperbolic flow, as in \cite{klw}, by using support function, we reduce it to a hyperbolic Monge-Amp$\grave{\rm{e}}$re equation …

New methods solve min-max problems on manifolds using Riemannian Hamiltonians.

New spacetimes found that are refocusing but not strongly refocusing.

The paper proves constant mean curvature surfaces in specific manifold types.

New group constructed from cube complex properties.

We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while retaining all of the equivalences to other notions of global hyperbolicity. In fact, global hyperbolicity is equivalent to the co…

Clarifies definitions of global hyperbolicity in various spaces.

Geometric operators link solutions on different spacetimes.

Graph products inherit Morse local-to-global property from their components.

Proves globally hyperbolic spacetimes via null distance completeness.

In this note a proof is given for global existence and uniqueness of minimal surfaces of Lorentzian type from a cylinder into globally hyperbolic Lorentzian manifolds for given initial values up to the first derivatives.

The paper solves the Cauchy problem for Friedrichs systems on specific spacetime manifolds.

This paper completes globally hyperbolic conformally flat spacetimes, proving they are topological manifolds.

Given a globally hyperbolic spacetime endowed with a complete lightlike Killing vector field and a complete Cauchy hypersurface, we characterize the points which can be connected by geodesics. A straightforward consequence is the geodesic connectedness of globally hyperbolic generalized plane waves with a complete Cauc…

Generalized canonical correlation analysis (GCCA) aims at finding latent low-dimensional common structure from multiple views (feature vectors in different domains) of the same entities. Unlike principal component analysis (PCA) that handles a single view, (G)CCA is able to integrate information from different feature …

An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic su…

We address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent scheme. With $O( μr^2 κ^2 n \max(μ, \log n))$ random observations of a $n_1 \times n…

Proves closure for specific spacetimes with certain conditions.

Minimal surfaces connect to horizons and electrostatic systems.

Synthetic proof shows globally hyperbolic Lorentzian spaces with specific curvature are warped products.

Deep neural networks have become commonplace in the domain of reinforcement learning, but are often expensive in terms of the number of parameters needed. While compressing deep neural networks has of late assumed great importance to overcome this drawback, little work has been done to address this problem in the conte…

It is shown that the space of null geodesics of a causally simple Lorentzian manifold is Hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime. This provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature con…

The chapter explores globally hyperbolic spacetimes using topology and functional analysis.

In this short note, a question of patching together globally hyperbolic manifolds is adressed which appeared in the context of the construction of Hadamard states.

Chernov-Nemirovski observed that the existence of a globally hyperbolic Lorentzian metric on a (3 + 1)-spacetime pins down a smooth structure on the underlying 4-manifold. In this paper, we point out that the diffeomorphism type of a globally hyperbolic (n + 1)-spacetime is determined by the h-cobordism class of its Ca…

While classic work in convex-concave min-max optimization relies on average-iterate convergence results, the emergence of nonconvex applications such as training Generative Adversarial Networks has led to renewed interest in last-iterate convergence guarantees. Proving last-iterate convergence is challenging because ma…

A new algorithm solves semidefinite programs using Langevin diffusion.

We give a topological condition for a generic sliced space to be globally hyperbolic, without any hypothesis on the lapse function, shift function and spatial metric.

Paper finds minimum volume for specific anti-de Sitter 3-manifolds.

A spacetime can be embedded in an enveloping space with all its extensions.

Globally hyperbolic spacetimes admitting infinitely many causal (and timelike) homotopy classes of curves joining two prescribed points, are exhibited and discussed.

We investigate 3-dimensional globally hyperbolic AdS manifolds containing "particles", i.e., cone singularities of angles less than $2π$ along a time-like graph $Γ$. To each such space we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities. We show that this data is sufficient …

We introduce the quasi-hyperbolicity constant of a metric space, a rough isometry invariant that measures how a metric space deviates from being Gromov hyperbolic. This number, for unbounded spaces, lies in the closed interval $[1,2]$. The quasi-hyperbolicity constant of an unbounded Gromov hyperbolic space is equal to…

Proposes an efficient alternative to nonconvex-nonconcave min-max optimization.