arXiv research

Proves unknottedness of certain 3D shapes with multiple ends.

We construct Gaussian Harmonic forms of finite Gaussian weighted $L^2$-norm on non-compact surfaces that detect each asymptotically conical end. As an application we prove an extension of the index estimates of self-shrinkers in $[11]$ under the existence of such ends. We show that the Morse index of a self-shrinker is…

Study on potential behavior in special geometric spaces.

Study conic singular manifolds, proving Lipschitz normal embedding.

Existence proof of noncompact self-shrinkers with arbitrary genus.

The paper proves a Penrose inequality for 4-discs with hyperbolic ends and conic singularities.

We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness…

Estimates ends of Ricci shrinkers, focusing on smooth and singular cases.

Paper proves finite Morse index for certain self-shrinkers.

The study proves Strichartz and spectral projection theorems on specific types of curved surfaces.

We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = Γ\backslash{\bf H}^2$ associated with a Fuchsian group of the 1st kind $Γ$ containing parabolic elements. $\mathcal M$ is t…

We show that if a shrinking soliton is asymptotic to a cone along an end then the isometry group of the cross-section of the cone embeds in the isometry group of the end of the shrinker. We also provide sufficient conditions for the isometries of the end to extend to the entire shrinker.

Using methods of A. Grigor'yan and L. Saloff-Coste we prove that on a manifold with a conical end the heat kernel has a Gaussian bound. This result is applied to asymptotically conical Kähler manifolds. It is a result of the author and R. Goto that a crepant resolution of a Ricci-flat Kähler cone admits a Ricci-flat Kä…

We prove two gluing theorems for special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. In particular, our theorems yield the first examples of smooth SL conifolds with 3 or more planar ends and the…

Uniqueness proven for specific types of geometric structures.

Let $C\subset\mathbb{R}^{n+1}$ be a regular cone with vertex at the origin. In this paper, we show the uniqueness for smooth properly embedded self-shrinking ends in $\mathbb{R}^{n+1}$ that are asymptotic to $C$. As an application, we prove that not every regular cone with vertex at the origin has a smooth complete pro…

Study on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.

We show that if two gradient Ricci solitons are asymptotic along some end of each to the same regular cone, then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does no…

If the initial hypersurface of an immortal mean curvature flow is asymptotic to a regular cone whose entropy is small, the flow will become asymptotically self-expanding. Moreover, the expander that gives rise to the limiting flow is asymptotically stable as an equilibrium solution of the normalized mean curvature flow…

We present a method to desingularize a compact G_2 manifold with isolated conical singularities by cutting out a neighbourhood of each singular point and glueing in an asymptotically conical G_2 manifold. Controlling the error on the overlap glueing region enables us to use a result of Joyce to conclude that the result…

We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we…

The study computes indicial roots and metric convergence orders for Ricci-flat conifolds.

The study examines asymptotic properties of G2-monopoles on nonparabolic G2-manifolds.

In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to $O(n)$-invariant co…

We analyze the resolvent $R(k)=(P+k^2)^{-1}$ of Schrödinger operators $P=Δ+V$ with short range potential $V$ on asymptotically conic manifolds $(M,g)$ (this setting includes asymptotically Euclidean manifolds) near $k=0$. We make the assumption that the dimension is greater or equal to 3 and that $P$ has no $L^2$ null …

New theorem shows noncompact self shrinkers are unknotted.

Desingularizes conically singular Cayley submanifolds.

Researchers create a parametrix for resolvents on manifolds with ends.

New self-expander found between two given asymptotic ones.

Proves mass theorem for AF manifolds with conical singularities.

Study instantons on asymptotically conical Spin(7)-manifolds, identifying deformation spaces.

Researchers found a family of Sp(2)-invariant solitons for Laplacian flow.

Study on how incomplete smooth metrics degenerate from asymptotically conical Ricci-flat Kähler metrics.

In many problems of PDE involving the Laplace-Beltrami operator on manifolds with ends, it is often useful to introduce radial or geodesic normal coordinates near infinity. In this paper, we prove the existence of such coordinates for a general class of manifolds with ends, which contains asymptotically conical and hyp…

The paper shows how to stabilize perturbed Kähler-Ricci solitons.

Proves positive mass theorem for AF spin manifolds with conical singularities.

Backwards uniqueness proved for flows with asymptotically conical singularities.

We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding ve…

The study shows that certain nearly G2 and nearly Kähler conifolds cannot be resolved by gluing asymptotically conical G2 and Calabi-Yau manifolds.

Ricci flow modelled on specific singularities on closed manifolds.

We discuss the deformation theory of special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. This category allows for the simultaneous presence of conical singularities and of non-compact, asymptotic…

Study of mean curvature flows with conical singularities using mathematical techniques.

Strict convexity proven for certain self-expanders in high dimensions.

We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.

Proves existence of Yamabe metrics on conical manifolds with conical points and links.

Backward propagation rules for warped products under Ricci flow.

In this paper we define a new convergence called "asymptotically conic convergence" in which a smooth family of Riemannian metrics on a fixed compact manifold degenerate to a metric with isolated conic singularity. Our results are: convergence of the spectrum of the geometric Laplacians and uniform convergence of the c…

Proves smoothness of conical singularities in mean curvature flow.