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In this work the Isoperimetric Inequality for integral varifolds is used to obtain sharp estimates for the size of the set where the density quotient is small and to generalise Calderons and Zygmunds theory of first order differentiability for functions in Lebesgue spaces from Lebesgue measure to integral varifolds.
We show that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging to a unio
We give several Bishop-Gromov relative volume comparisons with integral Ricci curvature which improve the results in cite{PW1}. Using one of these volume comparisons, we derive an estimate for the volume entropy in terms of integral Ricci curvature w
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 Weitzenbock type formula for critical metrics of the volume functio
In this work it is shown that every integral varifold in an open subset of Euclidian space of locally bounded first variation can be covered by a countable collection of submanifolds of class C^2. Moreover, the mean curvature of each member of the co
We prove an Herschs type isoperimetric inequality for the third positive eigenvalue on $mathbb S^2$. Our method builds on the theory we developped to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.