arXiv research

We find generators for the full rational loop group of GL(n,C) as well as for the subgroup consisting of loops that satisfy the reality condition with respect to the noncompact real form GL(n,R). We calculate the dressing action of some of those generators on the positive loop group, and apply this to the ZS-AKNS flows…

Projective loops generate rational loop groups without needing nilpotent loops.

An analysis is made of reality conditions within the context of noncommutative geometry. We show that if a covariant derivative satisfies a given left Leibniz rule then a right Leibniz rule is equivalent to the reality condition. We show also that the matrix which determines the reality condition must satisfy the Yang-…

It was proven by Hitchin that any solution of his evolution equations for a half-flat SU(3)-structure on a compact six-manifold M defines an extension of M to a seven-manifold with holonomy in G_2. We give a new proof, which does not require the compactness of M. More generally, we prove that the evolution of any half-…

We construct a local action of the group of rational maps from $S^2$ to $GL(n,C)$ on local solutions of flows of the ZS-AKNS $sl(n,C)$-hierarchy. We show that the actions of simple elements (linear fractional transformations) give local Bäcklund transformations, and we derive a permutability formula from different fact…

No expanding breathers found in noncompact Ricci flows with certain curvature conditions.

Study noncompact manifolds' Chern scalar curvatures, proving existence and multiplicity.

New theorem for spacetime mass in noncompact regions.

Study finds conditions for minimal surfaces in noncompact spaces.

Study gives conditions for noncompact manifolds to have positive scalar curvature.

Noncompact Ricci-flat solutions have infinite unstable dimensions.

Given a connected, compact, totally geodesic submanifold Y^m of noncompact type inside a compact locally symmetric space of noncompact type X^n, we provide a sufficient condition that ensures that [Y^m] is nonzero in H_m(X^n; R); in low dimensions, our condition is also necessary. We provide conditions under which ther…

Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on comp…

The paper proves a new inequality for 3-manifolds with noncompact boundaries.

Paper extends noncompact Llarull's theorem to manifolds with boundary.

Researchers prove Weyl laws for Schrödinger operators on noncompact manifolds.

We discuss some geometric conditions under which a complete noncompact shrinking gradient Ricci soliton will split at infinity.

Virtual reality explores non-Euclidean Sol geometry.

The study restricts scalar curvature on certain noncompact manifolds.

Since non-compact RCD(0, N) spaces have at least linear volume growth, we study noncompact RCD(0, N) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grow at most linearly on a noncompact RCD(0, N) space satisfying the linear volume growth…

In this paper, we consider the Yamabe equation on a complete noncompact Riemannian manifold and find some geometric conditions on the manifold such that the Yamabe problem admits a bounded positive solution.

We investigate complete noncompact Ricci-flat manifolds which are not of maximal volume growth. We show that the manifolds with a curvature decay condition and a holonomy decay condition are asymptotic to torus fibrations over ALE spaces. In particular, we classify complete noncompact 4-dimensional hyperkäler manifold…

Let $\M$ be a complete, connected noncompact manifold with bounded geometry. Under a condition near infinity, we prove that the Log Sobolev functional (\ref{logfanhan}) has an extremal function decaying exponentially near infinity. We also prove that an extremal function may not exist if the condition is violated. This…

Paper investigates prescribing Chern scalar curvatures on specific manifolds.

Study of $σ_k$ flow in Minkowski space for noncompact hypersurfaces.

Study calculates mass for certain hyperbolic spaces with noncompact boundaries.

We prove that a complete noncompact Kähler manifold $M^{n}$of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {\bf C}$^{n}$ and we show that the manifold is topologically {\bf R}$^{2n}$. In particular, when $M^{n}$ is a Kähler surface of positive bisecti…

We show that a totally geodesic submanifold of a symmetric space satisfying certain conditions admits an extension to a minimal submanifold of dimension one higher, and we apply this result to construct new examples of complete embedded minimal submanifolds in simply connected noncompact globally symmetric spaces.

LLM sandbox and persona dynamics create unethical reality gaps that shift risk to users.

No breathers found for mean curvature flow in noncompact space.

Study Liouville theorem for specific harmonic maps.

The paper proves conditions for positive scalar curvature and Yamabe constant on noncompact cylinders.

Study of prescribed mean curvature flow on noncompact hypersurfaces in Lorentz manifolds.

Paper proves existence of convex hypersurfaces with prescribed $p$-curvature.

We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a gen…

We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulations of three-dimensional hyperbolic space are available at http://h3.hypernom.com.

Survey examines challenges and solutions in sim-to-real transfer for robotics.

Morse theory extended to noncompact manifolds with complex geometric data.

Study local curvature estimates and existence of conformal metrics on noncompact manifolds.

We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In particular, for any noncompact covering of a compact manifold, there is a metric on the b…

Study Witten deformation on noncompact manifolds with bounded geometry.

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the uniformity of all estimates throughout the proof. The $C^{k,α}$-smoothness result is o…

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional curvature decay at most quadratically is diffeomorphic to a Euclidean n-space R^n und…

The aim of this paper is to study complete (noncompact) steady $m$-quasi-Einstein manifolds satisfying a fourth-order vanishing condition on the Weyl tensor. In this case, we are able to prove that a steady $m$-quasi-Einstein manifold ($m>1$) on a simply connected $n$-dimensional manifold $(M^{n},g)$, $(n\geq4),$ with …

Under suitable conditions near infinity and assuming boundedness of curvature tensor, we prove a no breathers theorem in the spirit of Ivey-Perelman for some noncompact Ricci flows. These include Ricci flows on asymptotically flat (AF) manifolds with positive scalar curvature. Since the method for the compact case face…

We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulation of the product of two-dimensional hyperbolic space with one-dimensional euclidean space is available at http://h2xe.hypernom.com.

VR enables professionals to develop deep learning models by moving virtual objects.

RG-2 flow reveals horizons in noncompact black hole metrics.