arXiv research

The paper shows that certain Einstein orbifolds cannot be limits of smooth Einstein metrics.

Study G-H limits of surfaces with boundary, focusing on same Euler characteristic.

We show that the Gromov-Hausdorff limit of a sequence of leaves in a compact foliation is a covering space of the limiting leaf which is no larger than this leaf's holonomy cover. We also show that convergence to such a limit is smooth instead of merely Gromov-Hausdorff. Corollaries include Reeb's local stability theor…

The paper examines sequences of metric spaces converging to compact limits with specific properties.

The paper examines convergence of distances in Lipschitz structures on manifolds.

Study resolvents of Bochner Laplacians on compact manifolds.

The paper studies limits of sub-Riemannian Heisenberg manifolds and proves convergence under certain conditions.

The study proves compactness and structure of Ricci flow limits.

Compact Ricci solitons on surfaces have at most two cone points, and are known as Hamilton's footballs. In this note we completely describe the degenerations of these footballs as one or both of the cone angles approaches zero. In particular, we show that Hamilton's famous non-compact cigar soliton is the Gromov--Hausd…

Computational method approximates homology groups of compact metric spaces.

This paper proves compactness of conformal Chern-minimal surfaces in Hermitian surfaces.

Survey of recent Kleinian representation convergence results.

Compactness theorem for Riemannian manifolds with volume and curvature bounds.

We obtain a Central Limit Theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin's Omnibus Central Limit Theorem for Fréchet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the ma…

In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe constants and…

We investigate variations of Brieskorn lattices over non-compact parameter spaces, and discuss the corresponding limit objects on the boundary divisor. We study the associated variation of twistors and the corresponding limit mixed twistor structures. We construct a compact classifying space for regular singular Briesk…

Smooth compactness theorem for elasticae, except straight segments.

Suppose $M$ is a complete, embedded minimal surface in $\mathbb{R}^3$ with an infinite number of ends, finite genus and compact boundary. We prove that the simple limit ends of $M$ have properly embedded representatives with compact boundary, genus zero and with constrained geometry. We use this result to show that if …

We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits $G/K = \varinjlim G_n/K_n$ of compact riemannian symmetric spaces, …

We study some similarities between almost product Riemannian structures and almost Hermitian structures. Inspired by the similarities, we prove lower eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds with locally product structures. We also provide some examples (limiting manifolds) for t…

The study examines gradient Ricci solitons with nonnegative curvature, proving properties of their blow-downs.

Study shows limits on deep and shallow neural networks for approximating compact sets.

We construct solutions to the constraint equations in general relativity using the limit equation criterion introduced by Dahl, Humbert and the first author. We focus on solutions over compact 3-manifolds admitting a $\bS^1$-symmetry group. When the quotient manifold has genus greater than 2, we obtain strong far from …

Paper shows limits of Heisenberg manifolds are flat tori.

The paper connects geodesic flows and limit sets on visibility manifolds.

Study shows normal distribution in divisor counts of random sections on complex manifolds.

We prove that there are compact submanifolds of the 3-sphere whose interiors are not homeomorphic to any geometric limit of hyperbolic knot complements.

No compact surfaces with specific curvature can exist near singular limits.

We construct a compact nonpositively curved squared 2-complex whose universal cover contains a flat plane that is not the limit of periodic flat planes.

We prove the existence of limits of real-analytic Laplace eigenvalue branches for real-analytic families of metrics that degenerate along a compact hypersurface.

We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination. We prove that if this limit projective lamination is doubly incompressi…

The paper characterizes $G$-ANR spaces and their properties.

In this note we establish several versions of a compactness theorem for submanifolds. In particular we require only bounds on the second fundamental form and do not assume volume or diameter bounds. As an application we prove a compactness theorem for mean curvature flows and use it to construct smooth blow-up limits a…

We provide a measure based topology for certain unions of C2 rectifiable submanifolds of mixed dimensions in Rn. In this topology lower dimensional sets remain in the limit as measures when higher dimensional sets collapse down to them. For example a decreasing sequence of spheres may have a limit consisting of just a …

In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a regular domain G of R^3. We prove that the second fundamental form of the boundary surface of G is nonnegatively defined at every point of the limit set of such immersions.

For convex co-compact hyperbolic manifolds $Γ\backslash \mathbb{H}^{n+1}$ for which the dimension of the limit set satisfies $δ_Γ< n/2$, we show that the high-frequency Eisenstein series associated to a point $ξ$ "at infinity" concentrate microlocally on a measure supported by (the closure of) the set of points in the …

The paper examines compactness of scalar curvature sequences on conformal manifolds.

For convex domains with $C^{1,ε}$ boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the automorphism group has finitely many components and the connected component of the identity is th…

In this paper, we analyze the asymptotic behaviour of the Hermitian-Yang-Mills flow over a compact non-Kähler manifold $(X,g)$ with the Hermitian metric $g$ satisfying the Gauduchon and Astheno-Kähler condition.

If $p : Y \to X$ is an unramified covering map between two compact oriented surfaces of genus at least two, then it is proved that the embedding map, corresponding to $p$, from the Teichmüller space ${\cal T}(X)$, for $X$, to ${\cal T}(Y)$ actually extends to an embedding between the Thurston compactification of the tw…

This paper studies limits of aspherical manifolds with specific curvature conditions.

In this article we study the limiting behavior of the Kähler Ricci flow on complete non-compact Kähler manifolds. We provide sufficient conditions under which a complete non-compact gradient Kähler-Ricci soliton is biholomorphic to $\ce^n$. We also discuss the uniformization conjecture by Yau \cite{Y} for complete non-…

Newly discovered Eguchi-Hanson metric arises from edge metrics.

We show that sequences of compact gradient Ricci solitons converge to complete orbifold gradient solitons, assuming constraints on volume, the $L^{n/2}$-norm of curvature, and the auxiliary constant $C_1$. The strongest results are in dimension 4, where $L^2$ curvature bounds are equivalent to upper bounds on the Euler…

We obtain an asymptotic formula for the spectrum distribution function of the Laplace operator on a compact Riemannian Sol-manifold in the adiabatic limit determined by a one-dimensional foliation defined by the orbits of a left-invariant flow.

Compactness theory for super Ricci flows provides convergence results.

Given a sequence of complete(compact or noncompact) Kähler manifolds $M^n_i$ with bisectional curvature lower bound and noncollapsed volume, we prove that the pointed Gromov-Hausdorff limit is homeomorphic to a normal complex analytic space. The complex analytic structure is the natural "limit" of complex structure of …

Haslhofer and Müller proved a compactness Theorem for four-dimensional shrinking gradient Ricci solitons, with the only assumption being that the entropy is uniformly bounded from below. However, the limit in their result could possibly be an orbifold Ricci shrinker. In this paper we prove a compactness theorem for non…