arXiv subjects

Almir Silva Santos

Publications and source records attributed to Almir Silva Santos.

14 recordsLinked to original sources

arXiv2026

Multiple Blow-Up Phenomena for $Q$-Curvature in High Dimensions

Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n \geq 25$ with positive Yamabe invariant $Y(M,g_0)>0$ and positive fourth-order invariant $Y_4(M,g_0)>0$. We show that, arbitrarily $C^1$-close to $g_0$, there exists a Riemannian metric such that, within its conformal class, one can find infinitely many smooth metrics with the same constant $Q$-curvature and arbitrarily large energy. Moreover, within this conformal class, there exists a sequence of smooth metrics with constant $Q$-curvature equal to $n(n^2-4)/8$ and unbounded volume. This extends to the $Q$-curvature setting the result previously obtained for the scalar curvature in Marques (2015) (see also Gond and Li (2025)). The proof is based on constructing small perturbations of multiple standard bubbles that are glued together.

math.DG↗

arXiv2025

An end to end gluing construction for metrics of constant Q-curvature

We produce many new complete, constant Q-curvature metrics on finitely punctured spheres by gluing together known examples. In our construction we truncate one end of each summand and glue the two summands together "end-to-end," where we've truncated them. We use this construction to show that the unmarked moduli space of solutions with a fixed number of punctures is topologically nontrivial provided the number of punctures is at least four.

math.DG↗

arXiv2025

Curvature Deformations on Complete Manifolds with Boundary

This paper investigates conformal deformations of the scalar curvature and mean curvature on complete Riemannian manifolds with boundary. We establish sufficient conditions for the existence of conformal deformations to complete metrics with positive scalar curvature and mean convex boundary. Building upon these results, we explore further deformation scenarios, including those that increase the mean curvature. Finally, we consider the case of deforming complete manifolds with negative scalar curvature on manifolds with noncompact boundary.

math.DG↗

arXiv2024

Blow-up analysis of Large conformal metrics with prescribed Gaussian and geodesic curvatures

Consider a compact Riemannian surface $(M,g)$ with nonempty boundary and negative Euler characteristic. Given two smooth non-constant functions $f$ in $M$ and $h$ in $\partial M$ with $\max f= \max h= 0$, under a suitable condition on the maximum points of $f$ and $h$, we prove that for sufficiently small positive constants $λ$ and $μ$, there exist at least two distinct conformal metrics $g_{λ,μ}=e^{2u_{μ,λ}}g$ and $g^{λ,μ}=e^{2u^{μ,λ}}g$ with prescribed sign-changing Gaussian and geodesic curvature equal to $f + μ$ and $h + λ,$ respectively. Additionally, we employ the method used in Borer et al. (2015) to study the blowing up behavior of the large solution $u^{μ,λ}$ when $μ\downarrow 0$ and $λ\downarrow 0$. Finally, we derive a new Liouville-type result for the half-space, eliminating one of the potential blow-up profiles.

math.DG↗

arXiv2024

Existence and obstructions for the curvature on compact manifolds with boundary

We study the set of curvature functions which a given compact manifold with boundary can possess. First, we prove that the sign demanded by the Gauss-Bonnet Theorem is a necessary and sufficient condition for a given function to be the geodesic curvature or the Gaussian curvature of some conformally equivalent metric. Our approach allows us to solve problems that are impossible to solve in the pointwise conformal case. Moreover, we obtain a deep and more delicate information on pointwise conformal deformations. We prove new existence and nonexistence results for metrics with prescribed curvature in the conformal setting, which depend on the Euler characteristic.

math.DG↗

arXiv2024

Moduli space theory for complete, constant Q-curvature metrics on finitely punctured spheres

We study constant Q-curvature metrics conformal to the round metric on the sphere with finitely many point singularities. We show that the moduli space of solutions with finitely many punctures in fixed positions, equipped with the Gromov-Hausdorff topology, has the local structure of a real analytic variety with formal dimension equal to the number of the punctures. If a nondegeneracy hypothesis holds, we show that a neighborhood in the moduli space is actually a real-analytic manifold of the expected dimension. We also construct a geometrically natural set of parameters, construct a symplectic structure on this parameter space and show that in the smooth case a small neighborhood of the moduli space embeds as a Lagrangian submanifold in the parameter space.

math.DG↗

arXiv2022

Constant Q-curvature metrics with Delaunay ends: the nondegenerate case

We construct a one-parameter family of solutions to the positive singular Q-curvature problem on compact nondegenerate manifolds of dimension bigger than four with finitely many punctures. If the dimension is at least eight we assume that the Weyl tensor vanishes to sufficiently high order at the singular points. On a technical level, we use perturbation methods and gluing techniques based on the mapping properties of the linearized operator both in a small ball around each singular point and in its exterior. Main difficulties in our construction include controlling the convergence rate of the Paneitz operator to the flat bi-Laplacian in conformal normal coordinates and matching the Cauchy data of the interior and exterior solutions; the latter difficulty arises from the lack of geometric Jacobi fields in the kernel of the linearized operator. We overcome both these difficulties by constructing suitable auxiliary functions.

math.DG↗

arXiv2022

On the $σ_2$-curvature and volume of compact manifolds

In this work we are interested in studying deformations of the $σ_2$-curvature and the volume. For closed manifolds, we relate critical points of the total $σ_2$-curvature functional to the $σ_2$-Einstein metrics and, as a consequence of results of H. J. Gursky and J. A. Viaclovsky (2001) and Z. Hu and H. Li (2004), we obtain a sufficient and necessary condition for a critical metric to be Einstein. Moreover, we show a volume comparison result for Einstein manifolds with respect to $σ_2$-curvature which shows that the volume can be controlled by the $σ_2$-curvature under certain conditions. Next, for compact manifold with nonempty boundary, we study variational properties of the volume functional restricted to the space of metrics with constant $σ_2$-curvature and with fixed induced metric on the boundary. We characterize the critical points to this functional as the solutions of an equation and show that in space forms they are geodesic balls. Studying second order properties of the volume functional we show that there is a variation for which geodesic balls are indeed local minimum in a natural direction.

math.DG↗

arXiv2020

Critical metrics and curvature of metrics with unit volume or unit area of the boundary

Given a smooth compact manifold with boundary, we study variational properties of the volume functional and of the area functional of the boundary, restricted to the space of the Riemannian metrics with prescribed curvature. We obtain a sufficient and necessary condition for a metric to be a critical point. As a by-product, a very natural analogue of V-statics metrics is obtained. In the second part, using the Yamabe invariant in the boundary setting, we solve the Kazdan-Warner-Kobayashi problem in a compact manifold with boundary. For several cases, depending on the signal of the Yamabe invariant, we give sufficient and necessary condition for a smooth function to be the scalar or mean curvature of a metric with constraint on the volume or area of the boundary.

math.DG↗

arXiv2020

Singular solutions to Yamabe-type systems with prescribed asymptotics

Our primary purpose is to study a class of strongly coupled nonlinear elliptic systems with critical growth in a compact Riemannian manifold with constant scalar curvature. Using a gluing technique and perturbation arguments, we show the existence of singular solutions asymptotic to a Fowler-type solution near the isolated singularity.

math.AP↗

arXiv2018

Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent

We studied the asymptotic behavior of local solutions for strongly coupled critical elliptic systems near an isolated singularity. For the dimension less than or equal to five we prove that any singular solution is asymptotic to a rotationally symmetric Fowler type solution. This result generalizes the celebrated work due to Caffarelli, Gidas, and Spruck [1] who studied asymptotic proprieties to the classic Yamabe equation. In addition, we generalize similar results by Marques [11] for inhomogeneous context, that is, when the metric is not necessarily conformally flat.

math.AP↗

arXiv2018

Deformation of the $σ_2$-curvature

Our main goal in this work is to deal with results concern to the $σ_2$-curvature. First we find a symmetric 2-tensor canonically associated to the $σ_2$-curvature and we present an Almost Schur Type Lemma. Using this tensor we introduce the notion of $σ_2$-singular space and under a certain hypothesis we prove a rigidity result. Also we deal with the relations between flat metrics and $σ_2$-curvature. With a suitable condition on the $σ_2$-curvature we show that a metric has to be flat if it is close to a flat metric. We conclude this paper by proving that the 3-dimensional torus does not admit a metric with constant scalar curvature and non-negative $σ_2$-curvature unless it is flat.

math.DG↗

arXiv2015

Solutions to the singular $σ_2-$Yamabe problem with isolated singularities

Given $(M,g_0)$ a closed Riemannian manifold and a nonempty closed subset $X$ in $M$, the singular $σ_k-$Yamabe problem asks for a complete metric $g$ on $M\backslash X$ conformal to $g_0$ with constant $σ_k-$curvature. The $σ_k-$curvature is defined as the $k-$th elementary symmetric function of the eigenvalues of the Schouten tensor of a Riemannian metric. The main goal of this paper is to find solutions to the singular $σ_2-$Yamabe problem with isolated singularities in any compact non-degenerate manifold such that the Weyl tensor vanishing to sufficiently high order at the singular point. We will use perturbation techniques and gluing methods.

math.DG↗

arXiv2009

A Construction of Constant Scalar Curvature Manifolds with Delaunay-type Ends

It has been showed by Byde that it is possible to attach a Delaunay-type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main goal of this paper is to generalize this result. We will construct a one-parameter family of solutions to the positive singular Yamabe problem for any compact non-degenerate manifold with Weyl tensor vanishing to sufficiently high order at the singular point. If the dimension is at most 5, no condition on the Weyl tensor is needed. We will use perturbation techniques and gluing methods.

math.DG↗