The Geometry Of Fractal Sets

Using min-max theory, we show that in any closed Riemannian manifold of dimension at least 3 and at most 7, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a conjecture of S.-T. Yau. This paper builds on the methods developed by F. C. Marques and A. Neves.

Existence of infinitely many minimal hypersurfaces in closed manifolds

In this paper, we prove that the Morse index of a multiplicity one, smooth, min-max minimal hypersurface is generically equal to the dimension of the homology class detected by the families used in the construction. This confirms part of the program (\cite{marques-icm}, \cite{marques-neves-cycles}, \cite{marques-neves-index}, \cite{neves-icm}) proposed by the authors with the goal of developing a Morse theory for the area functional.

Morse index of multiplicity one min-max minimal hypersurfaces

We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of $\mathbb{R}^n$. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarkably, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if all our weights are nonradial ---except for the constant ones. 
We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. 
As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella.

Sharp isoperimetric inequalities via the ABP method

In this paper, we develop a min–max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature c. Moreover, if c is nonzero then our min–max solution always has multiplicity one.

Min–max theory for constant mean curvature hypersurfaces

We completely characterize isoperimetric regions in R^n with density e^h, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.

Proof of the Log-Convex Density Conjecture

For a given null-cobordant Riemannian n
 n
 

-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on n
 n
 

. In the appendix the bound is improved to one that is O
 (
 L
 1
 +
 e
 )
 O(L^{1+\varepsilon })
 for every e
 >
 0
 \varepsilon >0
 

.

This construction relies on another of independent interest. Take X
 X
 and Y
 Y
 to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose Y
 Y
 is simply connected and rationally homotopy equivalent to a product of Eilenberg–MacLane spaces, for example, any simply connected Lie group. Then two homotopic L
 L
 

-Lipschitz maps f
 ,
 g
 :
 X
 →
 Y
 f,g:X \to Y
 are homotopic via a C
 L
 CL
 

-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces Y
 Y
 

.

/pdf/quantitative-null-cobordism-1ua9ettn9c.pdf

Quantitative null-cobordism

/pdf/isoperimetric-regions-in-rn-with-density-rp-25mtvwrz0d.pdf

Isoperimetric Regions in Rn with Density rp

We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume. The main tool is the following result of independent interest: if a region U can be swept out by a family of hypersurfaces of volume at most V, then it can be swept out by a family of mutually disjoint hypersurfaces of volume at most $$V + \varepsilon $$.

Gregory R. Chambers

Papers

Proof of the Log-Convex Density Conjecture

Quantitative null-cobordism

Isoperimetric Regions in Rn with Density rp

Existence of minimal hypersurfaces in complete manifolds of finite volume

Proof of the Log-Convex Density Conjecture