Geometry and Algebra Seminar

Abstract

We study critical points of functionals defined on certain linear spaces containing not functions, but k-dimensional geometric objects in ${\mathbf{R}}^n$ (e.g. varifolds or currents). The functionals of interest are defined as the Hausdorff measure with a weight depending on the tangent space, i.e., given a continuous positive real-valued function $F$ on the Grassmannian $\mathbf{G}(n,k)$ the associated functional $\mathrm{\Phi }$ on a smooth $k$-dimensional manifold $M$ yields an integral of $F(\mathrm{Tan}(M,\cdot ))$ with respect to the surface measure (Hausodrff measure) on $M$. Critical points of $\mathrm{\Phi }$ are defined with respect to one-parameter families of diffeomorphisms of $M$ which move points only in some compact set – formally: they are varifolds with zero first $F$-variation (a.k.a. $F$-stationary). In 2018, De Philippis, De Rosa and Ghirladin introduced the AC condition and showed that if the integrand $F$ satisfies it, then all $F$-stationary varifolds are rectifiable. In 2020, in a joint paper with De Rosa, we showed that the AC condition implies classical Almgren ellipticity for $\mathrm{\Phi }$. To date, both ellipticity and the AC condition have not been thoroughly studied. Aside from codimension one, no non-trivial (i.e., perturbations of the constant function) examples of functionals in the class AC are known. The talk will present a new look at the AC condition from the point of view of convex geometry.