Let $g_S$ be the Simanca metric on the blow-up $\tilde{\mathbb{C}}^2$ of $\mathbb{C}^2$ at the origin. We show that $(\tilde{\mathbb{C}}^2,g_S)$ admits a regular quantization. We use this fact to prove that all coefficients in the Tian-Yau-Zelditch expansion for the Simanca metric vanish and that a dense subset of $(\tilde{\mathbb{C}}^2, g_S)$ admits a Berezin quantization

The positive mass theorem states that the total mass of a complete asymptotically flat manifold with non-negative scalar curvature is non-negative; moreover, the total mass equals zero if and only if the manifold is isometric to the Euclidean space. Huang and Lee [2015] proved the stability of the Positive Mass Theorem for a class of $n$-dimensional ($n \geq 3$) asymptotically flat graphs with non-negative scalar curvature, in the sense of currents. Motivated by their work and using results of Dahl, Gicquaud and Sakovich [2013], we adapt their ideas to obtain a similar result regarding the stability of the positive mass theorem, in the sense of currents, for a class of $n$-dimensional $(n \geq 3)$ asymptotically hyperbolic graphs with scalar curvature bigger than or equal to $-n(n-1)$.

In this paper we prove existence and uniqueness of the Bjorling problem for the class of immersed surfaces in $$\mathbb {R}^3$$ whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with the topology of a Mobius strip for an arbitrary large class of prescribed functions. In particular, we use the Bjorling problem to construct the first known examples of self-translating solitons of the mean curvature flow with the topology of a Mobius strip in $$\mathbb {R}^3$$ .

We study the limiting behaviour of Darboux and Calapso transforms of polarized curves in the conformal n-dimensional sphere when the polarization has a pole of first or second order at some point. We prove that for a pole of first order, as the singularity is approached, all Darboux transforms converge to the original curve and all Calapso transforms converge. For a pole of second order, a generic Darboux transform converges to the original curve while a Calapso transform has a limit point or a limit circle, depending on the value of the transformation parameter. In particular, our results apply to Darboux and Calapso transforms of isothermic surfaces when a singular umbilic with index frac{1}{2} or 1 is approached along a curvature line.

We give a short proof of a strong version of the short-time asymptotic expansion of heat kernels associated with Laplace-type operators acting on sections of vector bundles over compact Riemannian manifolds, including exponential decay of the difference of the approximate heat kernel and the true heat kernel. We use this to show that repeated convolution of the approximate heat kernels can be used to approximate the heat kernel on all of M, which is related to expressing the heat kernel as a path integral. This scheme is then applied to obtain a short-time asymptotic expansion of the heat kernel at the cut locus.

Given a compact Riemannian manifold (M, g) and two positive functions $$\rho $$ and $$\sigma $$ , we are interested in the eigenvalues of the Dirichlet energy functional weighted by $$\sigma $$ , with respect to the $$L^2$$ inner product weighted by $$\rho $$ . Under some regularity conditions on $$\rho $$ and $$\sigma $$ , these eigenvalues are those of the operator $$ -\rho ^{-1} \text{ div }(\sigma \nabla u) $$ with Neumann conditions on the boundary if $$\partial M\ne \emptyset $$ . We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is preserved.

In this paper, we study the critical metrics for quadratic curvature functionals involving the Ricci curvature and scalar curvature in the space of Riemannian metrics with unit volume. For these functionals, Einstein metrics are always critical metrics. However, a converse problem is not always true. The purpose of this paper is to show that, under the condition that the critical metrics are Bach-flat, a partial converse is true.

We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The rôle of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this way, we recover many fundamental results in greater generality, while at the same time clarifying the minimal requirements for and the interdependence of the basic building blocks of the theory. A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT(k)-spaces, based on triangle comparison. Applications include Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity.

In this paper, we study stability properties of hypersurfaces with constant weighted mean curvature (CWMC) in gradient Ricci solitons. The CWMC hypersurfaces generalize the f-minimal hypersurfaces and appear naturally in the isoperimetric problems in smooth metric measure spaces. We obtain a result about the relationship between the properness and extrinsic volume growth under the assumption of a limitation for the weighted mean curvature of the immersion. Moreover, we estimate Morse index for CWMC hypersurfaces in terms of the dimension of the space of parallel vector fields restricted to hypersurface.

Let $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ be a K\"ahler manifold obtained by blowing up a complex projective space $\mathbb{P}^n$ along a line $\mathbb{P}^1$. We prove that $\text{Bl}_{\mathbb{P}^1} \mathbb{P}^n$ does not admit constant scalar curvature K\"ahler metrics in any rational K\"ahler class, but admits extremal metrics, with an explicit formula in action-angle coordinates, in K\"ahler classes that are close to the pullback of the Fubini--Study class.