scholarly journals Index theory for scalar curvature on manifolds with boundary

2021 ◽  
pp. 1
Author(s):  
John Lott
Keyword(s):  
Scalar Curvature ◽  
Index Theory ◽  
Manifolds With Boundary

scholarly journals A long neck principle for Riemannian spin manifolds with positive scalar curvature

2020 ◽  
Vol 30 (5) ◽  
pp. 1183-1223
Author(s):  
Simone Cecchini
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Pairwise Disjoint ◽  
Index Theory ◽  
Spin Manifold ◽  
Manifolds With Boundary ◽  
Topological Information ◽  
Manifold With Boundary ◽  
Spin Manifolds ◽  
Nonzero Degree

AbstractWe develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a “long neck principle” for a compact Riemannian spin n-manifold with boundary X, stating that if $${{\,\mathrm{scal}\,}}(X)\ge n(n-1)$$ scal ( X ) ≥ n ( n - 1 ) and there is a nonzero degree map into the sphere $$f:X\rightarrow S^n$$ f : X → S n which is strictly area decreasing, then the distance between the support of $$\text {d}f$$ d f and the boundary of X is at most $$\pi /n$$ π / n . This answers, in the spin setting and for strictly area decreasing maps, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold X obtained by removing k pairwise disjoint embedded n-balls from a closed spin n-manifold Y. We show that if $${{\,\mathrm{scal}\,}}(X)>\sigma >0$$ scal ( X ) > σ > 0 and Y satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of $$\partial X$$ ∂ X is at most $$\pi \sqrt{(n-1)/(n\sigma )}$$ π ( n - 1 ) / ( n σ ) . Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to $$N\times [-1,1]$$ N × [ - 1 , 1 ] , with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if $${{\,\mathrm{scal}\,}}(V)\ge \sigma >0$$ scal ( V ) ≥ σ > 0 , then the distance between the boundary components of V is at most $$2\pi \sqrt{(n-1)/(n\sigma )}$$ 2 π ( n - 1 ) / ( n σ ) . This last constant is sharp by an argument due to Gromov.

scholarly journals Positive scalar curvature and a new index theory for noncompact manifolds

2020 ◽  
Vol 149 ◽  
pp. 103575 ◽  
Author(s):  
Stanley Chang ◽  
Shmuel Weinberger ◽  
Guoliang Yu
Keyword(s):  
Scalar Curvature ◽  
Index Theory ◽  
Positive Scalar Curvature ◽  
Noncompact Manifolds ◽  
Positive Scalar

scholarly journals About the Uniqueness of Conformal Metrics with Prescribed Scalar and Mean Curvatures on Compact Manifolds with Boundary

2011 ◽  
Vol 13 ◽  
pp. 71-79
Author(s):  
Gonzalo García ◽  
Jhovanny Muñoz
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Manifolds With Boundary ◽  
Conformal Metrics ◽  
Dirichlet Eigenvalue ◽  
Uniqueness Results ◽  
Manifold With Boundary ◽  
First Dirichlet Eigenvalue ◽  
Neumann Eigenvalue ◽  
Linear Elliptic Problem

Let (Mn, g) be an n—dimensional compact Riemannian manifold with boundary with n > 2. In this paper we study the uniqueness of metrics in the conformai class of the metric g having the same scalar curvature in M, dM, and the same mean curvature on the boundary of M, dM. We prove the equivalence of some uniqueness results replacing the hypothesis on the first Neumann eigenvalue of a linear elliptic problem associated to the problem of conformai deformations of metrics for one about the first Dirichlet eigenvalue of that problem. Keywords: Conformal metrics, scalar curvature, mean curvature.

scholarly journals Width, Largeness and Index Theory

2020 ◽  
Author(s):  
Rudolf Zeidler ◽  
◽  
◽  
Keyword(s):  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Upper Bound ◽  
Index Theory ◽  
Positive Scalar Curvature ◽  
Point Of View ◽  
Open Problems ◽  
Spin Manifolds ◽  
Recent Developments ◽  
Theoretic Point

In this note, we review some recent developments related to metric aspects of scalar curvature from the point of view of index theory for Dirac operators. In particular, we revisit index-theoretic approaches to a conjecture of Gromov on the width of Riemannian bands M×[−1,1], and on a conjecture of Rosenberg and Stolz on the non-existence of complete positive scalar curvature metrics on M×R. We show that there is a more general geometric statement underlying both of them implying a quantitative negative upper bound on the infimum of the scalar curvature of a complete metric on M×R if the scalar curvature is positive in some neighborhood. We study (A hat-)iso-enlargeable spin manifolds and related notions of width for Riemannian manifolds from an index-theoretic point of view. Finally, we list some open problems arising in the interplay between index theory, largeness properties and width.

4-Dimensional manifolds with nonnegative scalar curvature and CMC boundary

2020 ◽  
pp. 1950094
Author(s):  
Yaohua Wang
Keyword(s):  
Scalar Curvature ◽  
Upper Bound ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Outer Boundary ◽  
Compact Manifolds ◽  
Manifolds With Boundary ◽  
Flat Extension ◽  
Asymptotically Flat ◽  
3 Dimensional

In this paper, we will consider 4-dimensional manifolds with nonnegative scalar curvature and constant mean curvature (CMC) boundary. For compact manifolds with boundary, the influence of the nonnegativity of the region scalar curvature to the geometry of the boundary is considered. Some inequalities are established for manifolds with inner boundary and outer boundary. Even for compact manifolds without inner boundary, we can obtain some inequalities involving the geometric quantities of the boundary and give some obstruction. We also discuss the 4-dimensional asymptotically flat extension of the 3-dimensional Bartnik data with CMC boundary and provide the upper bound of the Bartnik mass.

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