Mathematics

We prove that for a compact 3-manifold M with boundary admitting an ideal triangulation T with valence at least 10 at all edges, there exists a unique complete hyperbolic metric with totally geodesic boundary, so that T is isotopic to a geometric decomposition of M. Our approach is to use a variant of the combinatorial Ricci flow introduced by Luo [Luo05] for pseudo 3-manifolds. In this case, we prove that the extended Ricci flow converges to the hyperbolic metric exponentially fast.

In this paper, we adopt combinatorial Ricci curvature flow methods to study the existence of cusped hyperbolic structure on 3-manifolds with torus boundary. For general pseudo 3-manifolds, we prove the long-time existence and the uniqueness for the extended Ricci flow for decorated hyperbolic polyhedral metrics. We prove that the extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a decorated hyperbolic polyhedral metric of zero Ricci curvature. If it is the case, the flow converges exponentially fast. These results apply for cusped hyperbolic structure on 3-manifolds via ideal triangulation.

A hypersurface M in R n is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if the number of distinct principal curvatures is constant on M , i.e., each continuous principal curvature function has constant multiplicity on M . These conditions are preserved by stereographic projection, so this theory is essentially the same for hypersurfaces in R n or S n . The theory of compact proper Dupin hypersurfaces in S n is closely related to the theory of isoparametric hypersurfaces in S n , and many important results in this field concern relations between these two classes of hypersurfaces. In 1985, Cecil and Ryan conjectured on p. 184 of the book, "Tight and Taut Immersions of Manifolds," that every compact, connected proper Dupin hypersurface M??S n is equivalent to an isoparametric hypersurface in S n by a Lie sphere transformation. This paper gives a survey of progress on this conjecture and related developments.

Let M=G/H be a compact, simply connected, Riemannian homogeneous space, where G is (almost) effective and H is a simple Lie group. In this paper, we first classify all G -naturally reductive metrics on M , and then all G -geodesic orbit metrics on M .

In this paper we prove that the set of metrics conformal to the standard metric on S n ∖{ p 1 ,⋯, p l } is locally compact in C m,α topology for any m>0 , whenever the metrics have constant σ k curvature and the k -Dilational Pohozaev invariants have positive lower bound for k<n/2 . Here the k -Dilational Pohozaev invariants come from the Kazdan-Warner type identity for the σ k curvature, which is derived by Viaclovsky \cite{Viac2000} and Han \cite{H1}. When k=1 , Pollack \cite{Pollack} proved the compactness results for the complete metrics of constant positive scalar curvature on S n ∖{ p 1 ,⋯, p l } .

We develop a compactness theory for super Ricci flows, which lays the foundations for the partial regularity theory in [Bam20b]. Our results imply that any sequence of super Ricci flows of the same dimension that is pointed in an appropriate sense subsequentially converges to a certain type of synthetic flow, called a metric flow. We will study the geometric and analytic properties of this limiting flow, as well as the convergence in detail. We will also see that, under appropriate local curvature bounds, a limit of Ricci flows can be decomposed into a regular and singular part. The regular part can be endowed with a canonical structure of a Ricci flow spacetime and we have smooth convergence on a certain subset of the regular part.

In this paper we consider the moduli space of complete, conformally flat metrics on a sphere with k punctures having constant positive Q-curvature and positive scalar curvature. Previous work has shown that such metrics admit an asymptotic expansion near each puncture, allowing one to define an asymptotic necksize of each singular point. We prove that any set in the moduli space such that the distances between distinct punctures and the asymptotic necksizes all remain bounded away from zero is sequentially compact, mirroring a theorem of D. Pollack about singular Yamabe metrics. Along the way we define a radial Pohozaev invariant at each puncture and refine some a priori bounds of the conformal factor, which may be of independent interest.

We establish Sobolev type inequalities in the noncommutative settings by generalizing monotone metrics in the space of quantum states, such as matrix-valued Beckner inequalities. We also discuss examples such as random transpositions and Bernoulli-Laplace models.

We show that the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an arbitrary flat vector bundle has a meromorphic extension to the whole complex plane and that its leading term in the Laurent series at the zero point is related to the regularised determinant of the flat Laplacian of Cappell-Miller. When the flat vector bundle is close to an acyclic and unitary one, we show that the dynamical zeta function is regular at the zero point and that its value is equal to the complex valued analytic torsion of Cappell-Miller. This generalises author's previous results for unitarily flat vector bundles as well as Müller and Spilioti's results on hyperbolic manifolds.

We propose a novel meshless method to compute harmonic maps and conformal maps for surfaces embedded in the Euclidean 3-space, using point cloud data only. Given a surface, or a point cloud approximation, we simply use the standard cubic lattice to approximate its ϵ -neighborhood. Then the harmonic map of the surface can be approximated by discrete harmonic maps on lattices. The conformal map, or the surface uniformization, is achieved by minimizing the Dirichlet energy of the harmonic map while deforming the target surface of constant curvature. We propose algorithms and numerical examples for closed surfaces and topological disks.