arXiv subjects

Luis Atzin Franco Reyna

Publications and source records attributed to Luis Atzin Franco Reyna.

3 recordsLinked to original sources

arXiv2026

Entire area-minimizing surfaces in $\mathbf{R}^n$ of density two are quadratic or planar

We classify any entire area-minimizing surface $M^2\subset\mathbb{R}^n$ with density $2$ at infinity as either planar, or the zero set of a quadratic holomorphic polynomial inside some affine copy of $\mathbb{C}^2$. We also prove algebraicity of $M^2$ when the tangent cone at infinity has multiplicity one with general density. In our proof we introduce a sheeting-type theorem for minimizing surfaces near a multiplicity-two plane at infinity, and a new proof of holomorphicity (different from \cite{micallef}) based on a general ``calibrated at infinity'' principle.

math.DG↗

arXiv2026

Entire area-minimizing surfaces in R^4 are algebraic

We classify entire 2-dimensional area-minimizing or stable surfaces in R^4 with quadratic area growth as algebraic, cut out by a finite union of holomorphic polynomials whose collective degrees are controlled by the density at infinity. As a consequence, we obtain bounds on the singular set size and genus in terms of the density at infinity.

math.DG↗

arXiv2023

Decompositions of three-dimensional Alexandrov spaces

We extend basic results in $3$-manifold topology to general three-dimensional Alexandrov spaces (or Alexandrov $3$-spaces for short), providing a unified framework for manifold and non-manifold spaces. We generalize the connected sum to non-manifold $3$-spaces and prove a prime decomposition theorem, exhibit an infinite family of closed, prime non-manifold $3$-spaces which are not irreducible, and establish a conjecture of Mitsuishi and Yamaguchi on the structure of closed, simply-connected Alexandrov $3$-spaces with non-negative curvature. Additionally, we define a notion of generalized Dehn surgery for Alexandrov $3$-spaces and show that any closed Alexandrov $3$-space may be obtained by performing generalized Dehn surgery on a link in $S^3$ or the non-trivial $S^2$-bundle over $S^1$. As an application of this result, we show that every closed Alexandrov $3$-space is homeomorphic to the boundary of a $4$-dimensional Alexandrov space.

math.GT↗