N. S. Aiex

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20162026

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math.DG2026

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 $V^{n}$ in Riemannian manifolds $(N^{n + 1}, h)$ . Indeed, if…

math.DG2024

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.

math.DG2024

A Corrected Proof of the Graphical Representation of a Class of Curvature Varifolds by $C^{1, α}$ Multiple Valued Functions

Nicolau S. Aiex

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]. W…

math.DG2019

Index Estimates for Surfaces with Constant Mean Curvature in $3$ -dimensional Manifolds

Nicolau S. Aiex, Han Hong

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 a…

math.DG2018

Index Estimate of Self-Shrinkers in $R^{3}$ with Asymptotically Conical Ends

Nicolau Sarquis Aiex

We construct Gaussian Harmonic forms of finite Gaussian weighted $L^{2}$ -norm on non-compact surfaces that detect each asymptotically conical end. As an application we prove an exten…

math.DG2016

The Width of Ellipsoids

Nicolau Sarquis Aiex

We compute the $k$ -width of a round $2$ -sphere for $k = 1, \dots, 8$ and we use this result to show that unstable embedded closed geodesics can arise with multiplicity as a min-max cr…