arXiv subjects
Publications and source records attributed to Alessandro Carlotto.
Generalizing earlier work by Ros in ambient dimension three, we prove an affine lower bound for the Morse index of closed minimal hypersurfaces inside a flat torus in terms of their first Betti number (with purely dimensional coefficients).
We show that the Morse index of a properly embedded free boundary minimal hypersurface in a strictly mean convex domain of the Euclidean space grows linearly with the dimension of its first relative homology group (which is at least as big as the number of its boundary components, minus one). In ambient dimension three, this implies a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components. Thereby, the compactness theorem by Fraser and Li implies a strong compactness theorem for the space of free boundary minimal surfaces with uniformly bounded Morse index inside a convex domain. Our estimates also imply that the examples constructed, in the unit ball, by Fraser-Schoen and Folha-Pacard-Zolotareva have arbitrarily large index. Extensions of our results to more general settings (including various classes of positively curved Riemannian manifolds and other convexity assumptions) are discussed.
We show that asymptotically Schwarzschildean 3-manifolds cannot contain minimal surfaces obtained by perturbative deformations of a Euclidean catenoid, no matter how small the ADM mass of the ambient space and how large the neck of the catenoid itself. Such an obstruction is sharply three-dimensional and ceases to hold for more general classes of asymptotically flat data.
By extending and generalising previous work by Ros and Savo, we describe a method to show that the Morse index of every closed minimal hypersurface on certain positively curved ambient manifolds is bounded from below by a linear function of its first Betti number. The technique is flexible enough to prove that such a relation between the index and the topology of minimal hypersurfaces holds, for example, on all compact rank one symmetric spaces, on products of the circle with spheres of arbitrary dimension and on suitably pinched submanifolds of the Euclidean spaces. These results confirm a general conjecture due to Schoen and Marques-Neves for a wide class of ambient spaces.
We prove that if an asymptotically Schwarzschildean 3-manifold (M,g) contains a properly embedded stable minimal surface, then it is isometric to the Euclidean space. This implies, for instance, that in presence of a positive ADM mass any sequence of solutions to the Plateau problem with diverging boundaries can never have uniform height bounds, even at a single point. An analogous result holds true up to ambient dimension seven provided polynomial volume growth on the hypersurface is assumed.
In this article we investigate the restrictions imposed by the dominant energy condition (DEC) on the topology and conformal type of \textsl{possibly non-compact} marginally outer-trapped surfaces (thus extending Hawking's classical theorem on the topology of black holes). We first prove that an unbounded, stable marginally outer trapped surface in an initial data set $(M,g,k)$ obeying the dominant energy condition is conformally diffeomorphic to either the plane $\mathbb{C}$ or to the cylinder $\mathbb{A}$ and in the latter case infinitesimal rigidity holds. As a corollary, when the DEC holds strictly this rules out the existence of trapped regions with cylindrical boundary. In the second part of the article, we restrict our attention to asymptotically flat data $(M,g,k)$ and show that, in that setting, the existence of an unbounded, stable marginally outer trapped surface essentially never occurs unless in a very specific case, since it would force an isometric embedding of $(M,g,k)$ into the Minkowski spacetime as a space-like slice.
Given a compact 3-manifold N without boundary, we prove that for a bumpy metric of positive scalar curvature the space of minimal surfaces having a uniform upper bound on the Morse index is always finite unless the manifold itself contains an embedded minimal RP^2. In particular, we derive a generic finiteness result whenever N does not contain a copy of RP^3 in its prime decomposition. We discuss the obstructions to any further generalization of such a result. When the metric g is required to be (scalar positive and) strongly bumpy (meaning that all closed, immersed minimal surfaces do not have Jacobi fields, a notion recently proved to be generic by B. White) the same conclusion holds true for any closed 3-manifold.
The study of stable minimal surfaces in Riemannian $3$-manifolds $(M, g)$ with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when $(M, g)$ is asymptotically flat and has horizon boundary. As a consequence, we obtain an effective version of the positive mass theorem in terms of isoperimetric or, more generally, closed volume-preserving stable CMC surfaces that is appealing from both a physical and a purely geometric point of view. We also include a proof of the following conjecture of R. Schoen: An asymptotically flat Riemannian $3$-manifold with non-negative scalar curvature that contains an unbounded area-minimizing surface is isometric to flat $\mathbb{R}^3$.
We perform an optimal localization of asymptotically flat initial data sets and construct data that have positive ADM mass but are exactly trivial outside a cone of arbitrarily small aperture. The gluing scheme that we develop allows to produce a new class of $N$-body solutions for the Einstein equation, which patently exhibit the phenomenon of gravitational shielding: for any large $T$ we can engineer solutions where any two massive bodies do not interact at all for any time $t\in(0,T)$, in striking contrast with the Newtonian gravity scenario.
Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on the p-th Jacobi eigenvalue for p greater or equal than 2 one gains strong convergence to a smooth limit submanifold away from at most p-1 points.
We show that the Morse index of a closed minimal hypersurface in a four-dimensional Riemannian manifold cannot be bound in terms of the volume and the topological invariants of the hypersurface itself by presenting a method for constructing Riemannian metrics on S^4 that admit embedded minimal hyperspheres of uniformly bounded volume and arbitrarily large Morse index. The phenomena we exhibit are in striking contrast with the three-dimensional compactness results by Choi-Schoen.
We characterize the rate of convergence of a converging volume-normalized Yamabe flow in terms of Morse theoretic properties of the limiting metric. If the limiting metric is an integrable critical point for the Yamabe functional (for example, this holds when the critical point is non-degenerate), then we show that the flow converges exponentially fast. In general, we make use of a suitable Lojasiewicz-Simon inequality to prove that the slowest the flow will converge is polynomially. When the limit metric satisfies an Adams-Simon type condition we prove that there exist flows converging to it exactly at a polynomial rate. We conclude by constructing explicit examples to show that this does occur; these seem to be the first examples of a slowly converging solution to a geometric flow.
In this article we extend several foundational results of the theory of complete minimal surfaces of finite index in the Euclidean space to minimal surfaces in asymptotically flat manifolds and, more generally, to marginally outer-trapped surfaces in initial data sets of General Relativity. We show that if an asymptotically flat 3-manifold (M,g) of nonnegative scalar curvature contains a non-compact, properly embedded minimal surface which is stable and has quadratic area growth, then it is isometric to the flat R^{3}. This implies, for instance, that in presence of a positive ADM mass any sequence of solutions to the Plateau problem with diverging boundaries can never have uniform height bounds, even at a single point. The proof of this theorem is based on a characterization of finite index minimal surfaces, on classical infinitesimal rigidity results by Fischer-Colbrie and Schoen and on the positive mass theorem by Schoen-Yau. More specifically, we also show that a complete minimal surface of finite index inside an asymptotically flat 3-manifold has finitely many ends and each of these is a graph of a function that has a suitable expansion at infinity, in analogy with a classical result by Schoen for Euclidean spaces. In addition, we prove that a non-compact stable MOTS in an initial data set (M,g,k) is conformally diffeomorphic to either the plane C or to the cylinder A and in the latter case infinitesimal rigidity holds. If the data have harmonic asymptotics, the former case is also proven to be globally rigid in the sense that the presence of a stable MOTS forces an isometric embedding of (M,g,k) in the Minkowski space-time (\mathbb{M},η) as a space-like slice.
It was recently shown by R. Souam and E. Toubiana that the (non constantly curved) Berger spheres do not contain totally umbilic surfaces. Nevertheless in this article we show, by perturbative arguments, that all analytic metrics sufficiently close to the round metric $g_{0}$ on $\mathbb{S}^{3}$ possess \textsl{generalized} totally umbilic 2-spheres, namely critical points of the conformal Willmore functional $\int_Σ|A^\circ|^{2}\,dμ_γ$. The same is true in the smooth setting provided a suitable non-degeneracy condition on the traceless Ricci tensor holds. The proof involves a gluing process of two different finite-dimensional reduction schemes, a sharp asymptotic analysis of the functional on perturbed umbilic spheres of small radius and a quantitative Schur-type Lemma in order to treat the cases when the traceless Ricci tensor of the perturbation is degenerate but not identically zero. For left-invariant metrics on $SU(2)\cong \mathbb{S}^{3}$ our result implies the existence of uncountably many distinct Willmore spheres.
In the first part of this article, we complete the program announced in the preliminary note [8] by proving a conjecture presented in [9] that states the equivalence of contractibility and p_{1}-stability for generalized spaces of formal barycenters and hence we get purely algebraic conditions for the solvability of the singular Liouville equation on Riemann surfaces. This relies on a structure decomposition theorem for these model spaces in terms of maximal strata, and on elementary combinatorial arguments based on the selection rules that define such spaces. Moreover, we also show that these solvability conditions on the parameters are not only sufficient, but also necessary at least when for some $i\in{1,...,m}$ the value $α_{i}$ approaches -1. This disproves a conjecture made in Section 3 of [19] and gives the first non-existence result for this class of PDEs without any genus restriction. The argument we present is based on a combined use of maximum/comparison principle and of a Pohozaev type identity and applies for arbitrary choice both of the underlying metric g of $Σ$ and of the datum h.
Given a closed two dimensional manifold, we prove a general existence result for a class of elliptic PDEs with exponential nonlinearities and negative Dirac deltas on the right-hand side, extending a theory recently obtained for the regular case. This is done by global methods: since the associated Euler functional is in general unbounded from below, we need to define a new model space, generalizing the so-called space of formal barycenters and characterizing (up to homotopy equivalence) its very low sublevels. As a result, the analytic problem is reduced to a topological one concerning the contractibility of this model space. To this aim, we prove a new functional inequality in the spirit of [16] and then we employ a min-max scheme based on a cone-style construction, jointly with the blow-up analysis given in [5] (after [6] and [8]). This study is motivated by abelian Chern- Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature in presence of conical singularities (hence generalizing a problem raised by Kazdan and Warner in [26]).