Volume Estimates for Tubes Around Submanifolds Using Integral Curvature Bounds
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Volume Estimates for Tubes Around Submanifolds Using Integral Curvature Bounds Yousef K. Chahine1 Received: 8 November 2018 © Mathematica Josephina, Inc. 2019
Abstract We generalize an inequality of Heintze and Karcher (Ann Sci École Norm Sup 11(4):451–470, 1978) for the volume of tubes around minimal submanifolds to an inequality based on integral bounds for k-Ricci curvature. Even in the case of a pointwise bound, this generalizes the classical inequality by replacing a sectional curvature bound with a k-Ricci bound. This work is motivated by the estimates of Petersen– Shteingold–Wei (Geom Funct Anal 7(6):1011–1030, 1997) for the volume of tubes around a geodesic and generalizes their result. Using similar ideas we also prove a Hessian comparison theorem for k-Ricci curvature which generalizes the usual Hessian and Laplacian comparison for distance functions from a point and give several applications. Keywords Integral curvature bounds · Heintze–Karcher inequality · Intermediate Ricci curvature · Hessian comparison Mathematics Subject Classification Primary 53C20
1 Introduction The geodesic tube of radius r around a closed submanifold m of a Riemannian manifold M n , denoted T (, r ), is the set of all points whose distance to is at most r . In this paper, we give upper bounds for the volume of T (, r ) based on L p norms of the negative part of the k-Ricci curvature of M. For p = ∞, we prove that the well-known estimate of E. Heintze and H. Karcher based on pointwise sectional curvature bounds requires only k-Ricci bounds (Theorem 4.1). The main result is the case p < ∞, where we give the first estimates for the volume of tubes around submanifolds of general codimension using integral curvature bounds (Theorem 1.1).
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Yousef K. Chahine [email protected] University of California, Santa Barbara, USA
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Y. K. Chahine
The k-Ricci curvature interpolates between sectional curvature and Ricci curvature by taking an average of sectional curvatures over a k-dimensional subspace of the tangent space. Specifically, given a unit vector u tangent to M and k-dimensional subspace V of the tangent space orthogonal to u the k-Ricci curvature of (u, V) is defined by Rick (u, V) =
k
R(ei , u)u, ei ,
i=1
where e1 , . . . , ek form an orthonormal basis of V. Notice that Ricn−1 is equivalent to the Ricci curvature and Ric1 is equivalent to sectional curvature. We say that a manifold has k-Ricci curvature bounded below by k H for some constant H if Rick (u, V) ≥ k H for all unit vectors u ∈ T M and k-dimensional subspaces V ⊥ u. The earliest global results using k-Ricci lower bounds as a partial positivity condition for curvature were obtained by Wu [24], Shen [22], and Shen–Wei [21], though the relationship between k-Ricci curvature and volume had been considered previously by Bishop and Crittenden [2, p. 253]. A significant literature has since developed which bridges a gap between the global results based on sectional curvature bounds and those based on Ricci curvature bounds [7,10,11,17,23]. In a diff
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