arXiv research

Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, has at most one solution in the class of complete riemannian metric with complex se…

Lower bounds on curvature integral for manifolds with curvature constraints.

In this paper we give a new proof for an almost isometry theorem in Alexandrov spaces with curvature bounded below.

Sharp inequalities on curved spaces with bounded curvature.

A compactness theorem is proved for a family of Kähler surfaces with constant scalar curvature and volume bounded from below, diameter bounded from above, Ricci curvature bounded and the signature bounded from below. Furthermore, a splitting theorem and some rigidity theorems are proved for Einstein-Maxwell systems.

Using a method introduced by R. Bamler to study the behavior of scalar curvature under continuous deformations of Riemannian metrics, we prove that if a sequence of smooth Riemannian metrics gi on a fixed compact manifold M has isotropic curvature bounded from below by a nonnegative function u, and if gi converge in C …

We give a lower bound on the number of small positive eigenvalues of the p-form Laplacian in a certain type of collapse with curvature bounded below.

We prove the short-time existence of Ricci flows on complete manifolds with scalar curvature bounded below uniformly, Ricci curvature bounded below by a negative quadratic function, and with almost Euclidean isoperimetric inequality holds locally. In particular, this result applies to manifolds with both Ricci curvatur…

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.

Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded from below and $π:\bar{M}\to M$ be a normal, Riemannian cover. We show that, for any nonnegative function $f$ on $M$, the means of $føπ$ on the geodesic balls of $\bar{M}$ are comparable to the mean of $f$ on $M$. Combined with logarithmic volume est…

We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abe…

Introduces Alexandrov spaces with curvature below, covering various theorems.

We study non-collapsed Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below. Our main result is that each tangent cone is homeomorphic to a normal affine variety. This extends a result of Donaldson-Sun, who considered non-collapsed limits of polarized Kähler manifolds with two-sided Ricci curv…

We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.

We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows …

Estimates for Schrödinger operators on manifolds with bounded Ricci curvature.

We discuss in which sense general metric measure spaces possess a first order differential structure. Building on this, we then see that on spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting to define Hessian, covariant/exterior derivatives and Ricci curvature.

We prove the following results: An almost Hermitian manifold of indefinite metric is of pointwise constant holomorphic sectional curvature if the holomorphic sectional curvature is bounded from above and from below. If the antiholomorphic sectional curvature is bounded either from above or from below, then the manifold…

For Riemannian manifolds with a measure $(M,g, e^{-f} dvol_g)$ we prove mean curvature and volume comparison results when the $\infty$-Bakry-Emery Ricci tensor is bounded from below and $f$ is bounded or $\partial_r f$ is bounded from below, generalizing the classical ones (i.e. when $f$ is constant). This leads to ext…

Positive mass theorem for asymptotically flat manifolds with non-negative distributional scalar curvature

Eigenvalue estimates for Beltrami-Laplacian under specific curvature conditions.

We consider a complete biharmonic submanifold $φ:(M,g)\rightarrow (N,h)$ in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant $c$. Assume that the mean curvature is bounded from below by $\sqrt c$. If (i) $\int_M (|{\bf H}|^2-c)^{p}dv_g<\infty$, for some $0<p<\infty$, or (ii) …

The study sets limits on heat equation solutions' Hessians on curved spaces.

In this paper, we prove a new gradient estimate for minimal graphs defined on domains of a complete manifold with Ricci curvature bounded from below. In particular, we show that positive, entire minimal graphs on manifolds with non-negative Ricci curvature are constant, and that complete, parabolic manifolds with Ricci…

We show that non-collapsed Gromov-Hausdorff limits of polarized Kahler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson-Sun, in…

In this paper, an n-dimensional complete open manifold with nonnegative Ricci curvature and collapsing volume has been investigated. If its radial sectional curvature bounded from below, it shows that such a manifold is of finite topological type under some restrictions shown below.

We investigate complete minimal hypersurfaces in the Euclidean space $$, with Gauss-Kronecker curvature identically zero. We prove that, if $f:M^{3}\to {R}^{4}$ is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature b…

We prove global existence of Yamabe flows on non-compact manifolds $M$ of dimension $m\geq3$ under the assumption that the initial metric $g_0=u_0g_M$ is conformally equivalent to a complete background metric $g_M$ of bounded, non-positive scalar curvature and positive Yamabe invariant with conformal factor $u_0$ bound…

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of…

The study establishes conditions for orientability in spaces with lower Ricci curvature bounds.

Study collapsing geometry with Ricci curvature, proving Kähler metrics and Killing structures.

A comparison theorem for the isoperimetric profile on the universal cover of surfaces evolving by normalised Ricci flow is proven. For any initial metric, a model comparison is constructed that initially lies below the profile of the initial metric and which converges to the profile of the constant curvature metric. Th…

We recently established a Toponogov type triangle comparison theorem for a certain class of Finsler manifolds whose radial flag curvatures are bounded below by that of a von Mangoldt surface of revolution (arXiv:1205.3913). In this article, as its applications, we prove the finiteness of topological type and a diffeomo…

Let $\M$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. Associated with $L$ one has \textit{le carré du champ} $Γ$ and a canonical distance $d$, with respect to which we suppose that …

In this paper, we construct local and global solutions to the Kähler-Ricci flow from a non-collapsed Kähler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the Gromov-Hausdorff limit of sequence of complete noncompact non-collapsed Kähler manif…

In this paper, we explore the limit structure of a sequence of Riemannian manifolds with Bakry-Émery Ricci curvature bounded below in the Gromov-Hausdorff topology. By extending the techniques established by Cheeger-Cloding for Riemannian manifolds with Ricci curvature bounded below, we prove that each tangent space at…

The round sphere is stable among spin manifolds with a specific scalar curvature bound.

Study geodesic curvature in 2D Alexandrov spaces, generalizing results from spaces with curvature above.

In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvatur…

We introduce a new geometric invariant called the obtuse constant of spaces with curvature bounded below. We first find relations between this invariant and the normalized volume. We also discuss the case of maximal obtuse constant equal to $π/2$, where we prove some rigidity for spaces. Although we consider Alexandrov…

Let $M=X\times Y$ be the product of two complex manifolds of positive dimensions. In this paper, we prove that there is no complete Kähler metric $g$ on $M$ such that: either (i) the holomorphic bisectional curvature of $g$ is bounded by a negative constant and the Ricci curvature is bounded below by $-C(1+r^2)$ where …

Constructs metrics with zero eigenvalues for Hodge-Laplacian.

We establish several inequalities for manifolds with positive scalar curvature and, more generally, for the scalar curvature bounded from below, in the spirit of the classical bound on the distances between conjugates points in surfaces with positive sectional curvature.

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreo…

The paper finds and analyzes the Funk-Finsler structure in constant curvature spaces.

Assuming a lower bound on the Ricci curvature of a complete Riemannian manifold, for $p< 1/2$ we show the existence of bounds on the local $L^p$ norm of the Ricci curvature that depend only on the dimension and which improve with volume collapse.

I show that if a geodesic space has curvature bounded below locally in the sense of Alexandrov then its completion has the same lower curvature bound globally.

The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.