arXiv subjects

Eddygledson Souza Gama

Publications and source records attributed to Eddygledson Souza Gama.

5 recordsLinked to original sources

arXiv2026

Collapsed translating solitons with planar wings

We study complete embedded translating solitons in R^3 with finite genus and finite entropy. We prove that such collapsed translators whose wings are all of planar type (in a weak topological sense) must be vertical flat planes. In particular, this implies the non-existence result for gluing and doubling constructions involving planes that was conjectured in (14) and announced in (9) as an application of the authors' techniques, under weaker assumptions than previous rigidity results by Khan and Lynch-Tinaglia in (17,19,20). A key ingredient is a sharp new a priori decay estimate in the exterior problem for the horizontal graphical translator equation, showing that bounded solutions to the nonlinear equation, and their gradients, are controlled by multiples of the Green's function of the linear drift Laplacian, regardless of the size of the given L^\infty data. Combined with a graphical description of the infinite ends and the Neves-Tian flux identity, this yields the main rigidity result. As another application, we obtain a full classification of collapsed embedded finite-genus translators of entropy lambda < 3.

math.DG↗

arXiv2026

Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators

We show that mean curvature flow translators may exhibit non-removable singularities at infinity, due to jump discontinuities in their asymptotic profiles, and that oscillation can persist so as to yield a continuum of subsequential limit tangent planes. Nonetheless, we prove that as time $t\to\pm \infty$, any finite entropy, finite genus, embedded, collapsed translating soliton in $\mathbb{R}^3$ converges to a uniquely determined collection of planes. This requires global analysis of quasilinear soliton equations with non-perturbative drifts, which we analyze via sharp non-standard elliptic decay estimates for the drift Laplacian, implying improvements on the Evans-Spruck and Ecker-Huisken estimates in the soliton setting, and exploiting a link from potential theory of the Yukawa equation to heat flows with $L^\infty$-data on non-compact slice curves of these solitons. The structure theorem follows: such solitons decompose at infinity into standard regions asymptotic to planes or grim reaper cylinders. As one application, we classify collapsed translators of entropy two with empty limits as $t\to +\infty$.

math.DG↗

arXiv2025

Finite entropy translating solitons in slabs

We study translating solitons for the mean curvature flow, $Σ^2\subseteq\mathbb{R}^3$ which are contained in slabs, and are of finite genus and finite entropy. As a first consequence of our results, we can enumerate connected components of slices to define asymptotic invariants $ω^\pm(Σ)\in\mathbb{N}$, which count the numbers of "wings''. Analyzing these, we give a method for computing the entropies $λ(Σ)$ via a simple formula involving the wing numbers, which in particular shows that for this class of solitons the entropy is quantized into integer steps. Finally, combining the concept of wing numbers with Morse theory for minimal surfaces, we prove the uniqueness theorem that if $Σ$ is a complete embedded simply connected translating soliton contained in a slab with entropy $λ(Σ)=3$ and containing a vertical line, then $Σ$ is one of the translating pitchforks of Hoffman-Martín-White

math.DG↗

arXiv2023

Non-negative Ricci curvature and Minimal graphs with linear growth

We study minimal graphs with linear growth on complete manifolds $M^m$ with $\mathrm{Ric} \ge 0$. Under the further assumption that the $(m-2)$-th Ricci curvature in radial direction is bounded below by $C r(x)^{-2}$, we prove that any such graph, if non-constant, forces tangent cones at infinity of $M$ to split off a line. Note that $M$ is not required to have Euclidean volume growth. We also show that $M$ may not split off any line. Our result parallels that obtained by Cheeger, Colding and Minicozzi for harmonic functions. The core of the paper is a new refinement of Korevaar's gradient estimate for minimal graphs, together with heat equation techniques.

math.DG↗

arXiv2021

A barrier principle at infinity for varifolds with bounded mean curvature

Our work investigates varifolds $Σ\subset M$ in a Riemannian manifold, with arbitrary codimension and bounded mean curvature, contained in an open domain $Ω$. Under mild assumptions on the curvatures of $M$ and on $\partial Ω$, also allowing for certain singularities of $\partial Ω$, we prove a barrier principle at infinity, namely we show that the distance of $Σ$ to $\partial Ω$ is attained on $\partial Σ$. Our theorem is a consequence of sharp maximum principles at infinity on varifolds, of independent interest.

math.DG↗