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I don't know what a rectifiable varifold is, so I can't answer your questions rigorously. However, I doubt very much that there is any pathology in the plane. Also, using the principle that curvature …

Gross's log-Sobolev inequality is equivalent to Stam's inequality (which can be used to prove the Shannon's entropy power inequality). This inequality can in turn be interpreted as giving a sharp lowe …

If $(s,t) \mapsto \Gamma(s,t)$ is a family of curves in Riemannian manifold $M$, where $s \in [0,1]$ is the curve parameter and $t \in (-\delta,\delta)$ is the variation parameter, let $S = \partial_s …

This is really just a long commentary about your question. First, it is always possible to write everything without using coordinates, because the indices can refer to a (moving) frame of tangent vect …

One (but probably not the easiest) way to proceed is this: Just for the heck of it, I'll explain as much as I can in arbitrary dimensions and codimensions. Work with a fixed set of local co-ordinates …

I just remembered that there might another way using Jacobi fields. Suppose you have a hypersurface in an $(n+1)$-dimensional Riemannian manifold. You take advantage of the fact that along each geodes …

There is a theory for convex bodies. If $A, B \subset \mathbb{R}^n$ are convex bodies, whose interiors contain the origin, you can use set addition to define $A+tB$, for any $t \ge 0$, and $$ V(A,B) = …

Let $u: M\rightarrow N$ be a diffeomorphism. By staring at the Dirichlet energy formula and knowing that integration by parts works just as well here as for the classical case for functions, you know …

Although the answers to these regularity questions are not addressed explicitly in the paper cited in the comments to the question, they all follow from the formulas derived in that paper. 1) The ans …