We investigate the formation of singularities for surfaces evolving by volume preserving mean curvature flow. For axially symmetric flows—surfaces of revolution—in $\mathbb{R}^3$ with Neumann boundary conditions, we prove that the first developing singularity is of Type I. The result is obtained without any additional curvature assumptions being imposed, while axial symmetry and boundary conditions are justifiable given the volume constraint. Additional results and ingredients towards the main proof include a non-cylindrical parabolic maximum principle, and a series of estimates on geometric quantities involving gradient, curvature terms and derivatives thereof. These hold in arbitrary dimensions.
We study the propagator of the wave equation on a closed Riemannian manifold $M$. We propose a geometric approach to the construction of the propagator as a single oscillatory integral global both in space and in time with a distinguished complex-valued phase function. This enables us to provide a global invariant definition of the full symbol of the propagator—a scalar function on the cotangent bundle—and an algorithm for the explicit calculation of its homogeneous components. The central part of the paper is devoted to the detailed analysis of the subprincipal symbol; in particular, we derive its explicit small time asymptotic expansion. We present a general geometric construction that allows one to visualise obstructions due to caustics and describe their circumvention with the use of a complex-valued phase function. We illustrate the general framework with explicit examples in dimension two.
We give a solution of Plateau’s problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau’s problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger and hence works also in a quite general setting. However the main result of this paper seems to be new even in $\mathbb{R}^n$.
In this paper, we deal with stochastically complete submanifolds $M^n$ immersed with nonzero parallel mean curvature vector field in a Riemannian space form $\mathbb{Q}^{n+p}_c$ of constant sectional curvature $c \in {\lbrace -1, 0, 1 \rbrace}$. In this setting, we use the weak Omori–Yau maximum principle jointly with a suitable Simons type formula in order to show that either such a submanifold $M^n$ must be totally umbilical or it holds a sharp estimate for the norm of its total umbilicity tensor, with equality if and only if the submanifold is isometric to an open piece of a hyperbolic cylinder $\mathbb{H}^1 {\left( -\sqrt{1+r^2} \right)} \times \mathbb{S}^{n-1} (r)$ when $c=-1$, a circular cylinder $\mathbb{R} \times S^{n-1} (r)$, when $c=0$, and a Clifford torus $\mathbb{S}^1 {\left( 1-r^2 \right)} \times \mathbb{S}^{n-1} (r)$, when $c=0$.
We give an algorithm that, for a given value of the geometric genus $p_g$, computes all regular product-quotient surfaces with abelian group that have at most canonical singularities and have canonical system with at most isolated base points. We use it to show that there are exactly two families of such surfaces with canonical map of degree $32$. We also construct a surface with $q = 1$ and canonical map of degree $24$. These are regular surfaces with $p_g = 3$ and base point free canonical system. We discuss the case of regular surfaces with $p_g = 4$ and base point free canonical system.
In this paper, we classify all isomorphic classes of a family of Calabi–Yau $3$-folds with $20$ parameters. In addition, we show that the isomorphisms form a finite group. The invariants under the action of this group are calculated by introducing the so-called DS‑graph.
We show that Cayley graphs of virtually Abelian groups satisfy a Li–Yau type gradient estimate despite the fact that they do not satisfy any known variant of the curvature-dimension inequality with non-negative curvature.
Consider a compact Lie group $G$ and a closed subgroup $H \lt G$. Suppose $\mathcal{M}$ is the set of $G$-invariant Riemannian metrics on the homogeneous space $M = G/H$. We obtain a sufficient condition for the existence of $g \in \mathcal{M}$ and $c \gt 0$ such that the Ricci curvature of $g$ equals $cT$ for a given $T \in \mathcal{M}$. This condition is also necessary if the isotropy representation of $M$ splits into two inequivalent irreducible summands.
We prove the instability of some families of Riemannian manifolds with non-trivial real Killing spinors. These include the invariant Einstein metrics on the Aloff–Wallach spaces $N_{k,l} = \mathrm{SU}(3) / i_{k,l} (S^1)$ (which are all nearly parallel $\mathrm{G}_2$ except $N_{1,0}$), and Sasaki Einstein circle bundles over certain irreducible Hermitian symmetric spaces. We also prove the instability of most of the simply connected non-symmetric compact homogeneous Einstein spaces of dimensions $5$, $6$, and $7$, including the strict nearly Kähler ones (except $\mathrm{G}_2 / \mathrm{SU}(3)$).