Isoperimetric inequality and Weitzenböck type formula for critical metrics of the volume

Abstract

We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenböck type formula for critical metrics of the volume functional on four-dimensional manifolds. As an application, we obtain a classification result for such metrics. Date: December 1, 2017. 2010 Mathematics Subject Classification. Primary 53C25, 53C20, 53C21; Secondary 53C65. Key words and phrases. Volume functional; critical metrics; isoperimetric inequality; Weitzenböck formula. H. Baltazar was partially supported by CNPq/Brazil and FAPEPI/Brazil. E. Ribeiro Jr was partially supported by grants from CNPq/Brazil (Grant: 303091/2015-0), PRONEX-FUNCAP/CNPq/Brazil and CAPES/ Brazil -Finance Code 001.

Geometric inequalities for critical metrics of the volume functional

“…This result also holds for negative scalar curvature, provided that the mean curvature of the boundary satisfies H > 2, as was proven in [3]; see also [5]. Thereafter, Baltazar et al [2] were able to show an isoperimetric type inequality for Miao-Tam critical metrics with nonnegative scalar curvature. These type of estimates may be used to obtain new classification results as well as discard some possible new examples.…”

Geometry of compact quasi-Einstein manifolds with boundary

“…Among the contributions that motivated this work, we primarily mention the classical isoperimetric inequality and a result due to Shen [35] and Boucher, Gibbons and Horowitz [11] that asserts that the boundary ∂M of a compact three-dimensional oriented triple static space (i.e., 1-quasi-Einstein manifold) with connected boundary and scalar curvature 6 must be a 2-sphere whose area satisfies the inequality |∂M | ≤ 4π, with equality if and only if M 3 is equivalent to the standard hemisphere. In the same spirit, boundary estimates for V -static metrics and static spaces were established in, e.g., [1,3,6,7,20,21,24,29,32]. In the recent work [22, Theorem 1], Diógenes and Gadelha proved an analogous boundary estimate for compact m-quasi-Einstein manifolds M n with connected boundary ∂M by assuming the following conditions:…”

Integral and boundary estimates for critical metrics of the volume functional

“…This raised the question about the cases of negative and positive scalar curvature. In [3], Baltazar, Diógenes and Ribeiro established a sharp isoperimetric inequality for critical metrics with non-negative scalar curvature. In spite of that, the isoperimetric constant obtained by them depends on the potential function f. It would be interesting to see if such a constant can be improved to depend only on the dimension and mean curvature of the boundary.…”

“…As observed by Miao and Tam in [19], it is interesting to know whether the standard metrics on geodesic balls in space forms are the only critical metrics on simply connected compact manifolds with connected boundary. There have been a lot of advances concerning the rigidity of critical metrics of the volume functional; see, e.g., [3,4,5,6,7,12,16,19,20,27].…”

“…Besides being interesting on their own, such estimates play a fundamental role in proving classification results and discarding some possible new examples of special metrics on a given manifold. In recent years, it was established some useful boundary and volume estimates for critical metrics of the volume functional, as for example, isoperimetric and Shen-Boucher-Gibbons-Horowitz type inequalities (see, e.g., [1,3,6,7,12,14]). At the same time, it is natural to explore integral estimates in order to obtain new obstruction results.…”