arXiv subjects
Publications and source records attributed to Antoine Song.
One of S.T. Yau's problems asks the following: given a $3$-dimensional asymptotically flat manifold $M$ with non-negative scalar curvature and $L^2$-norm of the curvature tensor at most $1$, if the mass of $M$ is small, is there a bilipschitz diffeomorphism from $M$ to the flat Euclidean space $\mathbb{R}^3$? We provide a strong positive answer to this problem by using our previous work \cite{DS25}.
We prove that any random vector in $\mathbb{R}^n$ which is dominated in convex order by a standard Gaussian vector can be written as the sum of three standard Gaussian vectors. This implies that any $1$-subgaussian random vector in $\mathbb{R}^n$ is the sum of a universal number of Gaussian vectors. It also solves M. Talagrand's convexity problem, which in turn implies a weak version of a combinatorial analogue to the problem.
We prove that the convexity problem of M. Talagrand is equivalent to the subgaussian vector problem: can any centered $1$-subgaussian random vector in $\mathbb{R}^n$ be realized as the sum of a universal number of standard Gaussian vectors? We introduce methods to study this problem and, using elementary arguments, we settle it for $1$-subgaussian random variables and random vectors with good norm and covariance bounds. These results already confirm the permutation invariant case of the convexity problem, and give optimal estimates on the largest ellipsoid contained in a sum of large sets in Gaussian spaces. We also propose a Riemannian version of the convexity problem for spaces with nonnegative Ricci curvature.
Let $\tilde{\Sigma}$ be the universal cover of a closed surface $\Sigma$ of genus at least $2$. We characterize all equivariantly area-minimizing maps from $\tilde{\Sigma}$ to a Hilbert sphere, which are equivariant with respect to an isometric action of $\pi_1(\Sigma)$ weakly equivalent to the regular representation. As part of our proof, we classify all minimal surfaces in Hilbert spheres with constant negative Gaussian curvature. This builds on earlier results of E. Calabi, K. Kenmotsu, R. Bryant.
Let $M$ be a closed, oriented, negatively curved, $n$-dimensional manifold with fundamental group $\Gamma$. Let $S^\infty$ be the unit sphere in $\ell^2(\Gamma)$, on which $\Gamma$ acts by the regular representation. The spherical volume of $M$ is a topological invariant introduced by Besson-Courtois-Gallot. We show that it is equal to the area of an $n$-dimensional area-minimizing minimal surface inside the ultralimit of $S^\infty/\Gamma$, in the sense of Ambrosio-Kirchheim. Our proof combines the theory of metric currents with a study of limits of the regular representation of torsion-free hyperbolic groups.
Let $S$ be a punctured Riemann surface with Euler characteristic $\chi(S)<0$. For any unitary representation $\rho: \pi_1(S) \to U(N)$, we introduce its renormalized energy and its harmonic representatives, which are equivariant harmonic maps from the universal cover of $S$ to the unit sphere in $\mathbb{C}^N$. Our main result is that if a sequence of unitary representations $\rho_j$ strongly converges, then their renormalized energies converge to $\frac{\pi}{4}|\chi(S)|$ and the shape of their harmonic representatives converges to a unique rescaled hyperbolic metric. Combining this statement with examples of strongly converging representations provided by random matrix theory, we derive the following applications. (1) If $\pi_1(S)$ is a free group, then for a random $\rho: \pi_1(S) \to U(N)$, the shape of its harmonic representatives concentrates around a rescaled hyperbolic metric with high probability as $N\to \infty$. (2) For any closed hyperbolic surface, a finite covering admits a harmonic immersion into some Euclidean unit sphere, which is almost isometric after rescaling. (3) There are closed, branched, minimal surfaces $\mathfrak{S}_j$ in some Euclidean unit spheres such that $\mathfrak{S}_j$ Benjamini-Schramm converges to a rescaled hyperbolic plane as $j\to \infty$, and the Gaussian curvature $K_j$ of $\mathfrak{S}_j$ satisfies $\lim_{j\to \infty} \frac{1}{\mathrm{Area}(\mathfrak{S}_j)}\int_{\mathfrak{S}_j} |K_j+8|=0.$
We show that any closed hyperbolic 3-manifold M admits a Riemannian metric with scalar curvature at least -6, but with volume entropy strictly larger than 2. In particular, this construction gives counterexamples to a conjecture of I. Agol, P. Storm and W. Thurston.
We show that the Euclidean 3-space $\mathbb{R}^3$ is stable for the Positive Mass Theorem in the following sense. Let $(M_i,g_i)$ be a sequence of complete asymptotically flat $3$-manifolds with nonnegative scalar curvature and suppose that the ADM mass $m(g_i)$ of one end of $M_i$ converges to $0$. Then for all $i$, there is a subset $Z_i$ in $M_i$ such that $M_i\setminus Z_i$ contains the given end, the area of the boundary $\partial Z_i$ converges to zero, and $(M_i\setminus Z_i,g_i)$ converges to $\mathbb{R}^3$ in the pointed measured Gromov-Hausdorff topology for any choice of basepoints. This confirms a conjecture of G. Huisken and T. Ilmanen. Additionally, we find an almost quadratic upper bound for the area of $\partial Z_i$ in terms of $m(g_i)$. As an application of the main result, we also prove R. Bartnik's strict positivity conjecture.
Let $(M,g_0)$ be a closed oriented hyperbolic manifold of dimension at least $3$. By the volume entropy inequality of G. Besson, G. Courtois and S. Gallot, for any Riemannian metric $g$ on $M$ with same volume as $g_0$, its volume entropy $h(g)$ satisfies $h(g)\geq n-1$ with equality only when $g$ is isometric to $g_0$. We show that the hyperbolic metric $g_0$ is stable in the following sense: if $g_i$ is a sequence of Riemaniann metrics on $M$ of same volume as $g_0$ and if $h(g_i)$ converges to $n-1$, then there are smooth subsets $Z_i\subset M$ such that both $\mathrm{Vol}(Z_i,g_i)$ and $\mathrm{Area}(\partial Z_i,g_i)$ tend to $0$, and $(M\setminus Z_i,g_i)$ converges to $(M,g_0)$ in the measured Gromov-Hausdorff topology. The proof relies on showing that any spherical Plateau solution for $M$ is intrinsically isomorphic to $(M,\frac{(n-1)^2}{4n} g_0)$.
Given a closed oriented manifold or more generally a group homology class, we introduce the spherical Plateau problem, which is a variational problem corresponding to a topological invariant called the spherical volume. In principle, its solutions should be realized by minimal surfaces in quotients of spheres. We explain that in many geometrically interesting cases, those solutions are essentially unique. We start with a review of the Ambrosio-Kirchheim theory of metric currents, and the barycenter map method developed by Besson-Courtois-Gallot. Then, we outline the following applications: (1) the intrinsic uniqueness of spherical Plateau solutions for negatively curved, locally symmetric, closed oriented manifolds, (2) the intrinsic uniqueness of spherical Plateau solutions for all 3-dimensional closed oriented manifolds, (3) the construction of higher-dimensional analogues of hyperbolic Dehn fillings. We also propose some open questions.
Let $(M,g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2\leq n\leq 6$. Our main theorem generalizes the solution of S.-T. Yau's conjecture on the abundance of minimal surfaces and builds on a result of M. Gromov. Suppose that $(M,g)$ has bounded geometry, or more generally is thick at infinity. Then the following dichotomy holds for the space of closed hypersurfaces in $M$: either there are infinitely many saddle points of the $n$-volume functional, or there is none. Additionally, we give a new short proof of the existence of a finite volume minimal hypersurface in finite volume manifolds, we check Yau's conjecture for finite volume hyperbolic 3-manifolds and we extend the density result due to Irie-Marques-Neves when $(M,g)$ is shrinking to zero at infinity.
Let $M^{n+1}$ be a closed manifold of dimension $3\leq n+1\leq 7$. We show that for a $C^\infty$-generic metric $g$ on $M$, to any connected, closed, embedded, $2$-sided, stable, minimal hypersurface $S\subset (M,g)$ corresponds a sequence of closed, embedded, minimal hypersurfaces $\{Σ_k\}$ scarring along $S$, in the sense that the area and Morse index of $Σ_k$ both diverge to infinity and, when properly renormalized, $Σ_k$ converges to $S$ as varifolds. We also show that scarring of immersed minimal surfaces along stable surfaces occurs in most closed Riemannian $3$-manifods.
Let $R>1$ and let $B$ be the Euclidean $4$-ball of radius $R$ with a closed subset ${E}$ removed. Suppose that $B$ embeds symplectically into the unit cylinder $\mathbb{D}^2 \times \mathbb{R}^2$. By Gromov's non-squeezing theorem, ${E}$ must be non-empty. We prove that the Minkowski dimension of ${E}$ is at least $2$, and we exhibit an explicit example showing that this result is optimal at least for $R \leq \sqrt{2}$. In an appendix by Jo\'e Brendel, it is shown that the lower bound is optimal for $R < \sqrt{3}$. We also discuss the minimum volume of ${E}$ in the case that the symplectic embedding extends, with bounded Lipschitz constant, to the entire ball.
The minimal volume of a closed manifold $M$ is the infimum of the volume of $(M,g)$ over all metrics $g$ with sectional curvature between $-1$ and $1$. We introduce a variant called the essential minimal volume, $\mathrm{ess-Minvol}(M)$, which is the limit, as $\delta>0$ goes to $0$, of the infimum of the volume of the $\delta$-thick part of $(M,g)$ over all metrics $g$ with sectional curvature between $-1$ and $1$. We show that, for some universal constant $C>0$, any closed Einstein 4-manifold $M$ with Euler characteristic $e(M)$ satisfies $$C^{-1}e(M) \leq \mathrm{ess-Minvol}(M) \leq Ce(M).$$ As a corollary, these inequalities are true for the essential minimal volume of closed complex surfaces of nonnegative Kodaira dimension. We conjecture that those linear bounds in fact hold for the minimal volume.
We introduce a combinatorial argument to study closed minimal hypersurfaces of bounded area and high Morse index. Let $(M^{n+1},g)$ be a closed Riemannian manifold and $\Sigma\subset M$ be a closed embedded minimal hypersurface with area at most $A>0$ and with a singular set of Hausdorff dimension at most $n-7$. We show the following bounds: there is $C_A>0$ depending only on $n$, $g$, and $A$ so that $$\sum_{i=0}^n b^i(\Sigma) \leq C_A \big(1+index(\Sigma)\big) \quad \text{ if $3\leq n+1\leq 7$},$$ $$\mathcal{H}^{n-7}\big(Sing(\Sigma)\big) \leq C_A \big(1+index(\Sigma)\big)^{7/n} \quad \text{ if $n+1\geq 8$},$$ where $b^i$ denote the Betti numbers over any field, $\mathcal{H}^{n-7}$ is the $(n-7)$-dimensional Hausdorff measure and $Sing(\Sigma)$ is the singular set of $\Sigma$. In fact in dimension $n+1=3$, $C_A$ depends linearly on $A$. We list some open problems at the end of the paper.
Let $(M,g)$ be a closed oriented Riemannian $3$-manifold and suppose that there is a strongly irreducible Heegaard splitting $H$. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the stable oriented double cover of a non-orientable minimal surface with a vertical handle attached. In particular, this proves a result conjectured by Rubinstein. Some consequences include the existence in any $\mathbb{R}P^3$ of either a minimal torus or a minimal projective plane with stable universal cover. In the case of positive scalar curvature, it is shown for spherical space forms not diffeomorphic to $S^3$ or $\mathbb{R}P^3$ that any strongly irreducible Heegaard splitting admits a minimal representative in its isotopy class, and that there is a minimal Heegaard splitting of area less than $4π$ if $R\geq 6$.
Let $H$ be a strongly irreducible Heegaard surface in a closed oriented Riemannian $3$-manifold. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the boundary of a tubular neighborhood about a non-orientable minimal surface with a vertical handle attached. This confirms a long-standing conjecture of J. Pitts and J.H. Rubinstein.
For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in $M$. This gives a quantitative version of the main result of \cite{irie-marques-neves}, by Irie and the first two authors, that established denseness of minimal hypersurfaces for generic metrics. As in \cite{irie-marques-neves}, the main tool is the Weyl Law for the Volume Spectrum proven by Liokumovich and the first two authors in \cite{liokumovich-marques-neves}.