The isoperimetric profile of a smooth Riemannian manifold for small volumes

Abstract

We define a new class of submanifolds called pseudo-bubbles, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a Riemannian manifold, there is a unique family of concentric pseudo-bubbles which contains all the pseudo-bubbles C 2,α -close to small spheres. This permit us to reduce the isoperimetric problem for small volumes to a variational problem in finite dimension.

Extremal domains for the first eigenvalue in a general compact Riemannian manifold

“…Based on [25] and other related results, S. Narduli has obtained in [14] an asymptotic expansion of I τ as τ tends to 0. The parallel expansion of the Faber-Krahn profile is given in [2].…”

Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces

“…Indeed compact stable constant mean curvature hypersurfaces bounding a domain appear as solution of the isoperimetric problem. We know that solutions of this problem exist on any compact Riemannian manifold and are smooth possibly up to a singular set of codimension at least 8 (see theorem 1 of [18], see also [13] and [15]). Moreover, in any dimension, smooth solutions exist in a neighborhood of non-degenerate critical point of the scalar curvature ( [24]).…”

Some local eigenvalue estimates involving curvatures

“…To this end, thanks to the uniform bound on the L 2 norm of the Dirichlet energy in (12), it will suffice to show that no L 2 -mass is concentrated by w r at infinity, i.e. that for every δ > 0 there exists R > 0 such that sup r w r L 2 (R n+1 \R B) < δ.…”

“…As mentioned earlier Theorem 1.2 is a direct consequence of the expansion of the isoperimetric profile near zero for compact Riemannian manifolds that we recall here, see [6,12]. Letting…”

“…The second and third theorem are consequences of estimates of Cheeger isoperimetric profiles. To achieve this, we use the fine expansions of the isoperimetric profile near zero by Druet [6] and Narduli [12] in compact Riemannian manifolds and the relative isoperimetric profile near zero by the author in [7], see Sect. 4.…”