Width estimate and doubly warped product

@article{Zhu2020WidthEA,
  title={Width estimate and doubly warped product},
  author={Jintian Zhu},
  journal={arXiv: Differential Geometry},
  year={2020}
}
  • Jintian Zhu
  • Published 3 March 2020
  • Mathematics
  • arXiv: Differential Geometry
In this paper, we give an affirmative answer to Gromov's conjecture ([3, Conjecture E]) by establishing an optimal Lipschitz lower bound for a class of smooth functions on orientable open $3$-manifolds with uniformly positive sectional curvatures. For rigidity we show that the universal covering of the given manifold must be $\mathbf R^2\times (-c,c)$ with some doubly warped product metric if the optimal bound is attained. This gives a characterization for doubly warped product metrics with… 
View PDF on arXiv
19 Citations
Highly Influential Citations
4
Background Citations
8
Methods Citations
6

19 Citations

Rigidity results for complete manifolds with nonnegative scalar curvature

In this paper, we are going to show some rigidity results for complete open Riemannian manifolds with nonnegative scalar curvature. Without using the famous Cheeger-Gromoll splitting theorem we give…

Scalar and mean curvature comparison via $\mu$-bubbles

. Following ideas of Gromov we prove scalar and mean curvature comparison results for Riemannian bands with lower scalar curvature bounds in dimension n ≤ 7. The model spaces we use are warped…

Gehring Link Problem, Focal Radius and Over-torical width

In this note, we study the Gehring link problem in the round sphere, which motives our study of the width of a band in positively curved manifolds. Using the same idea, we are able to get a sphere…

Generalized soap bubbles and the topology of manifolds with positive scalar curvature

We prove that for $n\in \{4,5\}$, a closed aspherical $n$-manifold does not admit a Riemannian metric with positive scalar curvature. Additionally, we show that for $n\leq 7$, the connected sum of a…

Scalar and mean curvature comparison via the Dirac operator

We use the Dirac operator technique to establish sharp distance estimates for compact spin manifolds under lower bounds on the scalar curvature in the interior and on the mean curvature of the…

The Gauss–Bonnet inequality beyond aspherical conjecture

. Up to dimension five, we can prove that given any closed Riemannian manifold with nonnegative scalar curvature, of which the universal covering has vanishing homology group H k for all k ≥ 3 ,…

Topology of $3$-manifolds with uniformly positive scalar curvature

  • 2022

Dimension Constraints in Some Problems Involving Intermediate Curvature

In [2] Brendle-Hirsch-Johne proved that T m × S n − m does not admit metrics with positive m -intermediate curvature when n ≤ 7. Chu-Kwong-Lee showed in [4] a corresponding rigidity statement when n…

A Generalization of the Geroch Conjecture with Arbitrary Ends

. Using µ -bubbles, we prove that for 3 ≤ n ≤ 7, the connected sum of a Schoen-Yau-Schick n -manifold with an arbitrary manifold does not admit a complete metric of positive scalar curvature. When…

Spectral Torical Band Inequalities and Generalizations of the Schoen-Yau Black Hole Existence Theorem

Generalized torical band inequalities give precise upper bounds for the width of compact manifolds with boundary in terms of positive pointwise lower bounds for scalar curvature, assuming certain…

References

SHOWING 1-10 OF 11 REFERENCES

Rigidity of area-minimizing $2$-spheres in $n$-manifolds with positive scalar curvature

  • Jintian Zhu
  • Mathematics
    Proceedings of the American Mathematical Society
  • 2020
We prove that the least area of the non-contractible immersed spheres is no more than $4\pi$ in any oriented compact manifold with dimension $n+2\leq 7$ which satisfies $R\geq 2$ and admits a map to…

Rigidity of min-max minimal spheres in three-manifolds

In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or…

A Splitting Theorem for Scalar Curvature

We show that a Riemannian 3‐manifold with nonnegative scalar curvature is flat if it contains an area‐minimizing cylinder. This scalar‐curvature analogue of the classical splitting theorem of J.…

Metric Inequalities with Scalar Curvature

  • M. Gromov
  • Mathematics
    Geometric and Functional Analysis
  • 2018
We establish several inequalities for manifolds with positive scalar curvature and, more generally, for the scalar curvature bounded from below. In so far as geometry is concerned these inequalities…

Existence of hypersurfaces with prescribed mean curvature I – generic min-max

We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The…

One-sided complete stable minimal surfaces

We prove that there are no complete one-sided stable minimal surfaces in the Euclidean 3-space. We classify least area surfaces in the quotient of R by one or two linearly independent translations…

Rigidity of area-minimizing two-spheres in three-manifolds

We give a sharp upper bound for the area of a minimal two-sphere in a three-manifold (M,g) with positive scalar curvature. If equality holds, we show that the universal cover of (M,g) is isometric to…

Can We Make a Finsler Metric Complete by a Trivial Projective Change

A trivial projective change of a Finsler metric F is the Finsler metric F + d f. I explain when it is possible to make a given Finsler metric both forward and backward complete by a trivial…

Four Lectures on Scalar Curvature

We overview main topics and ideas in spaces with their scalar curvatures bounded from below, and present a more detailed exposition of several known and some new geometric constraints on Riemannian…

Local Rigidity Theorems for Minimal Hypersurfaces