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Banach spaces have a relatively uninteresting topology, because they are contractible. This prevents the direct application of Morse-like min-max arguments to establish the existence of critical point …

I am interested in calculating the least area of a surface spanning the boundary of an octant on the unit sphere; and short of precise values I am looking for upper bounds for this area. In $\mathbf{S …

Question. My question in broad terms: when and how can entropy be used in geometric variational contexts to study sequences of critical points, such as the two results described below? [See at the bot …

Let $T$ be an $n$-dimensional area-minimising hypersurface in $\mathbf{R}^{n+1}$. If $T$ has bounded area growth in the sense that there is a constant $C > 0$ so that $\mathcal{H}^n(T \cap B_R) \leq C …

Let $U \subset \mathbf{R}^{n+k}$ be a bounded open set, and $T \in \mathbf{I}_n(U)$ be an $n$-dimensional integral rectifiable current. Say that $T$ is stationary through (homological) deformations if …

Let $n \geq 2$, and let $M^n \subset \mathbf{R}^{n+1}$ be a minimal surface with $0 \in M$ and finite ($n$-dimensional) area growth: $\operatorname{limsup}_{R \to \infty} R^{-n} \lVert M \cap B_R \rVe …

I am reading a paper of Comte and Mironescu [CM96], where they discuss critical points $v = v_{\epsilon}: G \to \mathbf{C}$ of the (non-magnetic) Ginzburg–Landau functional $E_\epsilon(v) = \frac{1}{2 …

Let $\mathbf{C}^n \subset \mathbf{R}^{n+1}$ be a minimizing cone with an isolated singularity. One example, in a space of even dimension, i.e. if $\mathbf{R}^{n+1} = \mathbf{R}^{2m}$, is the Simons co …

Let me state my question in very loose terms to start, then give some details and restate it in more precise terms at the bottom. Question. Given a regular minimal cone $\mathbf{C}$, can one set up a …

Let $M^{n-1}$ be a smooth closed manifold, embedded into the round sphere $\mathbf{S}^n$ via a regular map $\Phi$. Using tools from geometric measure theory, one can prove the existence of a $n$-dimen …

Let $M^8 \subset B^9 \setminus \{ 0 \} \subset \mathbf{R}^9$ be a properly embedded, stable minimal hypersurface. Suppose that $0 \in \overline{M}$ is an isolated singularity of the surface. Question. …

I recently stumbled across a quote of Fang-Hua Lin that I have trouble understanding [1, page 42]. It is a well-known fact that a weakly converging sequence of stationary integral currents may have a …