arXiv research

Study shows certain surface homeomorphisms groups can't be precompact.

Einstein metrics on homogeneous torus bundles

New principles prove precompactness of domains with lower Ricci curvature bound.

The paper defines a new concept of approximability for Lagrangian submanifolds.

Ancient Ricci flows on compact spaces converge to solitons.

There are only a few invariants one classically associates with precompact translation surfaces, among them certain numberfields, i.e. fields which are finite extensions of the field Q of rational numbers. These fields are closely related to each other; they are often even equal. We prove by constructing explicit examp…

This note proves the geodesic completeness of any compact manifold endowed with a linear connection such that the closure of its holonomy group is compact.

New approach to prescribing Gaussian curvature on spheres with conical singularities.

This survey reviews precompactness theorems for classes of Riemannian manifolds with boundary. We begin with the works of Kodani, Anderson-Katsuda-Kurylev-Lassas-Taylor and Wong. We then present new results of Knox and the author with Sormani.

Compact metrics found for Riemannian manifolds with controlled curvature.

In this short paper, we will study the convergence of a metric sequence $g_k$ on $S^2$ with bounded $\int|K_{g_k}|\log(1+|K_{g_k}|)dμ_{g_k}$ and small $\int|K(g_k)-1\|dμ_{g_k}$. We will show that such a sequence is precompact.

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the loca…

We show precompactness results for solutions to parabolic fourth order geometric evolution equations. As part of the proof we obtain smoothing estimates for these flows in the presence of a curvature bound, an improvement on prior results which also require a Sobolev constant bound. As consequences of these results we …

We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's $ν$-entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the $κ$-noncollapsing property. Finally, we us…

In this paper we study global distance estimates and uniform local volume estimates in a large class of sub-Riemannian manifolds. Our main device is the generalized curvature dimension inequality introduced by the first and the third author in \cite{BG1} and its use to obtain sharp inequalities for solutions of the sub…

We establish an integral formula on a smooth, precompact domain in a Kahler manifold. We apply this formula to study holomorphic extension of CR functions. Using this formula we prove an isoperimetric inequality in terms of a positive lower bound for the Hermitian curvature of the boundary. Combining with a Minkowski t…

We study the motion of an $n$-dimensional closed spacelike hypersurface in a Lorentzian manifold in the direction of its past directed normal vector, where the speed equals a positive power $p$ of the mean curvature. We prove that for any $p\in (0,1]$, the flow exists for all time when the Ricci tensor of the ambient s…

Compactness results for Hermitian manifolds help understand Type IIB flow.

We consider a geometric property of the closest-points projection to a geodesic in Teichmüller space: the projection is called contracting if arbitrarily large balls away from the geodesic project to sets of bounded diameter. (This property always holds in negatively curved spaces.) It is shown here to hold if and only…

New Dynkin condition for manifolds with boundary yields bi-Lipschitz equivalence and spectral properties.

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a th…

In this paper we provide an extension to the Jellett-Minkowski's formula for immersed submanifolds into ambient manifolds which possesses a pole and radial curvatures bounded from above or below by the radial sectional curvatures of a rotationally symmetric model space. Using this Jellett-Minkowski's generalized formul…

In this paper, we prove $W^{1,p}$ ($p>n$) and $C^{0,α}$ ($0 < α< 1$) precompactness for classes of Riemannian $n$-manifolds with boundary satisfying uniform $L^{\infty}$ bounds on curvature, mean curvature, diameter, and the $(n-1)$-volume of the boundary. In particular, we identify a class of convex manifolds and a cl…

The paper examines torsional rigidity bounds under geometric flows.

We present a geometric analysis of the incompressible averaged Euler equations for an ideal inviscid fluid. We show that solutions of these equations are geodesics on the volume-preserving diffeomorphism group of a new weak right invariant pseudo metric. We prove that for precompact open subsets of ${\mathbb R}^n$, thi…

The paper constructs Gromov-Hausdorff metrics for Lorentzian spaces and calculates dimensions.

Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.

Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(M_i,g_i(t), O_i)} such that t in [0,T] are solutions to the Ricci flow and g_i(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then t…

Extends compactness theory to variable-coefficient pseudo-differential operators on manifolds.

In this paper, we study contact structures on any open 3-manifold V which is the interior of a compact 3-manifold. To do this, we introduce proper contact isotopy invariants called the slope at infinity and the division number at infinity. We first prove several classification theorems for T^2 x [0, \infty), T^2 x R, a…

Paper reconciles different Ricci flow approaches and proves weak solutions.

This paper concerns complete noncompact manifolds with nonnegative Ricci curvature. Roughly, we say that M has the loops to infinity property if given any noncontractible closed curve, C, and given any compact set, K, there exists a closed curve contained in M\K which is homotopic to C. The main theorems in this paper …

We prove that if a family of metrics, $g_i$, on a compact Riemannian manifold, $M^n$, have a uniform lower Ricci curvature bound and converge to $g_\infty$ smoothly away from a singular set, $S$, with Hausdorff measure, $H^{n-1}(S) = 0$, and if there exists connected precompact exhaustion, $W_j$, of $M^n \setminus S$ s…

Constructs solutions of Einstein equations for black holes gluing along timelike geodesics.

Study heat flow on collapsing K3 surfaces, handling conic singularities.