[Paper Review] Continuity and differentiability properties of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry

Abraham Muñoz Flores, Stefano Nardulli|arXiv (Cornell University)|Apr 11, 2014

Geometric Analysis and Curvature Flows13 references13 citations

TL;DR

This paper establishes the continuity and differentiability properties of the isoperimetric profile $I_M$ in complete noncompact Riemannian manifolds with bounded geometry—defined by Ricci curvature bounded below and uniformly positive volume of unit balls. Under additional geometric constraints, it proves $I_M$ satisfies a differential inequality and extends classical results from compact to noncompact settings.

ABSTRACT

For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is continuous. Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below and volume of balls of radius $1$, uniformly bounded below with respect to its centers. Then under an extra hypothesis on the geometry of $M$, we apply this result to prove some differentiability property of $I_M$ and a differential inequality satisfied by $I_M$, extending in this way well known results for compact manifolds, to this class of noncompact complete Riemannian manifolds with bounded geometry.

Motivation & Objective

• To establish the continuity of the isoperimetric profile $I_M$ in complete noncompact Riemannian manifolds with bounded geometry.
• To extend differentiability and differential inequality results for $I_M$ from compact to noncompact manifolds under additional geometric hypotheses.
• To show equivalence between strong and weak formulations of the isoperimetric profile using finite perimeter sets.
• To investigate whether continuity and concavity-type properties of $I_M$ persist under Ricci curvature lower bounds and potential collapsing.
• To explore the validity of Bavard-Pansu-type inequalities in noncompact, non-collapsing settings.

Proposed method

• Uses the weak formulation of the isoperimetric profile via finite perimeter sets, defined as $\tilde{I}_M(v) = \inf\{\mathcal{P}(\Omega) : V(\Omega) = v, \Omega \in \tilde{\tau}_M\}$, to generalize the classical definition.
• Applies the theory of functions of bounded variation (BV) and reduced boundaries $\partial^*\Omega$ to handle nonsmooth sets and ensure geometric regularity.
• Employs comparison geometry via the model space $\mathbb{M}_k^n$ with constant curvature $k$ to bound mean curvature and inradius of isoperimetric regions.
• Uses the noncollapsing condition (uniform positive volume of unit balls) to construct test sets by removing or adding small geodesic balls to prove semicontinuity.
• Applies the generalized isoperimetric inequality from Bavard-Pansu (Appendix) to derive a differential inequality for $I_M - Cv^2$ in the sense of distributions.
• Relies on the existence of isoperimetric regions for all volumes $v \in (0, V(M))$ to derive second-order distributional bounds on $I_M$.

Experimental results

Research questions

• RQ1 Does the isoperimetric profile $I_M$ remain continuous in complete noncompact Riemannian manifolds with bounded geometry?
• RQ2 Under what geometric conditions can differentiability and differential inequalities for $I_M$ be extended from compact to noncompact manifolds?
• RQ3 Are the strong and weak formulations of the isoperimetric profile equivalent in noncompact manifolds?
• RQ4 Can the concavity-type inequality from Bavard-Pansu be generalized to noncompact, non-collapsing manifolds with Ricci curvature bounded below?
• RQ5 Is continuity of $I_M$ guaranteed under Ricci lower bound and existence of isoperimetric regions, even in collapsing settings?

Key findings

• The isoperimetric profile $I_M$ is continuous on $[0, V(M))$ for any complete noncompact Riemannian manifold with bounded geometry, defined by $\mathrm{Ric} \geq k$ and $\mathrm{Vol}(B(p,1)) \geq v_0 > 0$ for all $p \in M$.
• Under the existence of isoperimetric regions for all volumes $v \in (0, V(M))$, the function $v \mapsto I_M(v) - C v^2$ has nonpositive second derivative in the sense of distributions, for some constant $C = C(a,b,n,k,M)$.
• The differential inequality $I_M''(v) \leq -\frac{(n-1)k}{I_M(v)}$ holds in the distributional sense when $k < 0$, under the same existence assumption.
• The weak and strong formulations of the isoperimetric profile are equivalent: $I_M = \tilde{I}_M$, even in noncompact settings.
• The proof of continuity relies on constructing test sets by adding or removing small geodesic balls of uniform size, exploiting the noncollapsing condition.
• The result extends classical results from compact manifolds to noncompact ones under bounded geometry, and the authors leave open whether continuity holds under Ricci lower bound alone without noncollapsing.