arXiv subjects
Publications and source records attributed to Nicolau S. Aiex.
We prove that all reasonable definitions of multiple-valued functions with continuous derivatives are equivalent, which eliminates inconsistencies in the literature. As an intermediate result we prove that affinely approximable functions defined on intervals admit a differentiable selection.
We establish quantitative topological and singularity properties for (certain) prescribed mean curvature (PMC) hypersurfaces $V^n$ in Riemannian manifolds $(N^{n+1},h)$. Indeed, if $V$ has area at most $A>0$ with PMC given by a $C^{1,α}$ function $g:N\to \mathbb{R}$ with the bound $|g|_{C^{1,α}}\leq Γ$, we show that there exists a constant $C$ depending only on $n,h,A,Γ$ and geometric quantities such that: \[\sum^n_{i=0}b^i(V) \leq C(1+\text{index}(V)) \quad \text{if }3\leq n+1\leq 7;\] \[M^{*n-7}(\text{sing}(V)) \leq C(1+\text{index}(V)) \quad \text{if }n+1\geq 8.\] Here, $b^i$ denote the Betti numbers over any field, $M^{*n-7}$ denotes the upper $(n-7)$-dimensional Minkowski content, and $\text{sing}(V)$ is the singular set of $V$. The first inequality extends the work of Song from the minimal hypersurface setting to the PMC hypersurface setting, whilst the second extends work of the authors. Our results apply to the PMC hypersurfaces constructed recently through min-max techniques by Bellettini--Wickramasekera.
We construct a curvature varifold that does not admit a decomposition whose components are curvature varifolds.
We prove that the space of free boundary CMC surfaces of bounded topology, bounded area and bounded boundary length is compact in the $C^k$ graphical sense away from a finite set of points. This is a CMC version of a result for minimal surfaces by Fraser-Li \cite{fraser.a-li.m2014}.
We provide a counter-example to Hutchinson's original proof of $C^{1,α}$ representation of curvature $m$-varifolds with $L^q$-integrable second fundamental form and $q>m$ in [6]. We also provide an alternative proof of the same result and introduce a method of decomposing varifolds into nested components preserving weakly differentiability of a given function. Furthermore, we prove the structure theorem for curvature varifolds with null second fundamental form which is widely used in the literature.
We prove local measure bounds on the tubular neighbourhood of the singular set of codimension one stationary integral $n$-varifolds $V$ in Riemannian manifolds which have both: (i) finite index on their smoothly embedded part; and (ii) $\mathcal{H}^{n-1}$-null singular set. A direct consequence of such a bound is a bound on the upper Minkowski content of the singular set of such a varifold in terms of its total mass and its index. Such a result improves on known bounds, namely the corresponding bound on the $\mathcal{H}^{n-7}$-measure of the singular set established by A. Song ([10]), as well as the same bounds established by A. Naber and D. Valtorta ([6]) for codimension one area minimising currents. Our results also provide more structural information on the singular set for codimension one integral varifolds with finite index (on the regular part) and no classical singularities established by N. Wickramasekera ([11]).
We prove index estimates for closed and free boundary CMC surfaces in certain $3$-dimensional submanifolds of some Euclidean space. When the mean curvature is large enough we are able to prove that the index of a CMC surface in an arbitrary $3$-manifold is bounded below by a linear function of its genus.