10 papers
On Almgren's Multiple-valued Functions of Class 1
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 intermed…
Quantitative Estimates on the Topology and Singular Set of Prescribed Mean Curvature Hypersurfaces
Nicolau S. Aiex, Sean McCurdy, Paul Minter
We establish quantitative topological and singularity properties for (certain) prescribed mean curvature (PMC) hypersurfaces in Riemannian manifolds . Indeed, if…
A Curvature Varifold Whose Weak Second Fundamental Form is Not Preserved Under Decompositions
Nicolau S. Aiex
We construct a curvature varifold that does not admit a decomposition whose components are curvature varifolds.
A Corrected Proof of the Graphical Representation of a Class of Curvature Varifolds by Multiple Valued Functions
Nicolau S. Aiex
We provide a counter-example to Hutchinson's original proof of representation of curvature -varifolds with -integrable second fundamental form and in [6]. W…
Quantitative Estimates on the Singular Set of Minimal Hypersurfaces with Bounded Index
Nicolau S. Aiex, Sean McCurdy, Paul Minter
We prove local measure bounds on the tubular neighbourhood of the singular set of codimension one stationary integral -varifolds in Riemannian manifolds which have both: (i)…
Compactness of the Space of Free Boundary CMC Surfaces with Bounded Topology
Nicolau S. Aiex, Han Hong
We prove that the space of free boundary CMC surfaces of bounded topology, bounded area and bounded boundary length is compact in the graphical sense away from a finite set o…
Index Estimates for Surfaces with Constant Mean Curvature in -dimensional Manifolds
Nicolau S. Aiex, Han Hong
We prove index estimates for closed and free boundary CMC surfaces in certain -dimensional submanifolds of some Euclidean space. When the mean curvature is large enough we are a…
Index Estimate of Self-Shrinkers in with Asymptotically Conical Ends
Nicolau Sarquis Aiex
We construct Gaussian Harmonic forms of finite Gaussian weighted -norm on non-compact surfaces that detect each asymptotically conical end. As an application we prove an exten…
The Width of Ellipsoids
Nicolau Sarquis Aiex
We compute the -width of a round -sphere for and we use this result to show that unstable embedded closed geodesics can arise with multiplicity as a min-max cr…
Non-compactness of the space of minimal hypersurfaces
Nicolau Sarquis Aiex
We show that the space of min-max minimal hypersurfaces is non-compact when the manifold has an analytic metric of positive Ricci curvature and dimension . Further…