arXiv research

Sharp estimate shown to be rigid on curved surfaces.

Proves nonnegatively curved hypersurfaces in hyperbolic space are usually properly embedded.

Study on spaces of nonnegatively curved surfaces and their topological properties.

Rigidity theorem for curved surfaces, proving uniqueness up to motion.

We give an explicit estimate of the area of a closed surface by the diameter and a lower bound of curvature. This is better than Calabi-Cao's estimate for a nonnegatively curved two-sphere.

Estimates submanifold diameters in curved spaces.

We provide new results and new proofs of results about the torsion of curves in $\mathbb{R}^3$. Let $γ$ be a smooth curve in $\mathbb{R}^3$ that is the graph over a simple closed curve in $\mathbb{R}^2$ with positive curvature. We give a new proof that if $γ$ has nonnegative (or nonpositive) torsion, then $γ$ has zero …

We provide techniques for studying the nonnegatively curved left-invariant metrics on a compact Lie group. For "straight" paths of left-invariant metrics starting at bi-invariant metrics and ending at nonnegatively curved metrics, we deduce a nonnegativity property of the initial derivative of curvature. We apply this …

Totally geodesic dual leaves on curved manifolds are also curved.

Let V be an open manifold with complete nonnegatively curved metric such that the normal sphere bundle to a soul has no section. We prove that the souls of nearby nonnegatively curved metrics on V are smoothly close. Combining this result with some topological properties of pseudoisotopies we show that for many V the s…

The study counts geodesics on curved surfaces with specific intersections.

Proves nonnegatively curved hypersurfaces on a sphere are convex disks.

Eigenvalues bound between Laplace and Steklov on curved surfaces.

Study shows unbounded Pontryagin numbers on curved manifolds.

Study on curved spheres' topology.

Study shows nonnegatively curved 7D manifolds with specific properties.

Strict convexity is essential for compact minimal surfaces in curved spaces.

New examples show nonnegatively curved metrics with same soul but different moduli space components.

We study the J-flow on Kahler surfaces when the Kahler class lies on the boundary of the open cone for which global smooth convergence holds, and satisfies a nonnegativity condition. We obtain a C^0 estimate and show that the J-flow converges smoothly to a singular Kahler metric away from a finite number of curves of n…

Study shows infinitely many nonnegatively curved metric components on Milnor spheres.

Given compact Lie groups H\subset G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a …

We establish a link between rational homotopy theory and the problem which vector bundles admit complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if C lies in the class and T is a torus of positive dimensi…

The paper shows Morse-Novikov cohomology vanishes for certain curved manifolds.

We show that almost nonnegatively curved m-dimensional manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m)-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanif…

We prove that a class of asymptotically nonnegatively curved manifolds (in the sense of Abresch) satisfying some uniform Euclidean type volume growth conditions contains only finitely many homeomorphism types.

We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we classify all knots in S^3 which can be realized as an extremal set with respect to an …

Study involutions on spaces to compute nonnegatively curved metrics dimensions.

Minimal graph level sets are strictly convex in curved spaces.

Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.

Study proves inequalities for mass-capacity on curved spaces.

We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of ``bounded geometry''.

We study nonnegatively curved metrics on S^2xR^4. First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking's almost-positively curved metric on S2xS3 extends to a nonnegatively curved metr…

Study growth rates of harmonic functions on curved surfaces.

This is a slightly altered version of the authors thesis from 2014. In the first main part we show that the quotient space of a compact, simply connected and nonnegatively curved Riemannian 4-manifold by an effective, isometric circle-action admits an approximation in Gromov-Hausdorff topology by smooth, positively cur…

Study shows infinitely many nonnegatively curved metric spaces on exotic 7-manifolds.

The study classifies singularities in curved 3D shapes.

Study shows infinitely many nonnegatively curved metrics on quotient spaces.

Sharp inequality found for hypersurfaces in curved spaces.

Proves conjecture about foliations on curved spaces.

We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metric.

We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension $4k \geq 12$, there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we raise and analyze a question about possible cobordism obstructions to non…

B. Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature, and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We…

Classifies Euclidean submanifolds with nonnegative curvature in high dimensions.

Study shows fundamental groups of certain curved spaces are simple.

In this paper, we study a complete noncompact nonnegatively curved Alexandrov space $A$ with a soul $S$ of codimension two. We establish some structural results under additional regularity assumptions. As an application, we conclude that in this case Sharafutdinov retraction, $π:\ A\to S$, is a submetry.

We classify the complete metrics of nonnegative sectional curvature on M^2xR^2, where M^2 is any compact 2-manifold.

In curved spaces, isoperimetric sets don't exist for small volumes.

We apply various topological methods to distinguish connected components of moduli spaces of complete Riemannian metrics of nonnegative sectional curvature on open manifolds. The new geometric ingredient is that souls of nearby nonnegatively curved metrics are ambiently isotopic.