Progetti di ricerca annuali e biennali

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1 SNS - Ricerca di base - Programma Luciano Mari Progetti di ricerca annuali e biennali Luciano Mari February 18, 2018 Abstract The present project focuses on two topics. The first one concerns isoperimetric inequalities on Cartan-Hadamard manifolds, that is, on complete, simply connected Riemannian manifolds with non-positive sectional curvature. A typical example is the Euclidean space R n, where the isoperimetric inequality reads as follows: iso_rn (1) vol(ω) n 1 n S n vol( Ω) Ω open, smooth, relatively compact, for some explicit constant S n, with equality holding if and only if Ω is an Euclidean ball (that is, S n = vol(b 1 ) n 1 n /vol( B 1 ) in R n ). Inequality (1) with the same constant also holds on the hyperbolic space H n k of sectional curvature k2, irrespectively of the value of k (cf. [3]). However, in H n k there exists no mimimizer for (1). A natural question is whether (1) holds on each Cartan- Hadamard manifold, a large class of relevant Riemannian spaces. Give a positive answer to this question is known as the Cartan-Hadamard conjecture, a classical problem addressed by mathematicians at least since the eighties, see [6] for applications. Explicit S n realizing (1) were found for any n, but the conjectured sharp value of S n, that is, the one of the Euclidean space, is only known for n = 2, 3, 4, see [3, 9, 5]. Interestingly, the proofs in dimensions 2, 3, 4 are different each other and heavily depend on the dimension restriction. A generalization of the Cartan-Hadamard conjecture is the isoperimetric conjecture for minimal submanifolds, stating the validity of (1) with the Euclidean constant whenever M n is a complete, proper minimal submanifold of some Cartan-Hadamard ambient space, typically R N. In view of [12, 8, 2] (1) holds for some explicit S n, but only in dimension n = 2 the constant S n is sharp (cf. [3]). We aim to address the Cartan- Hadamard conjecture for any n, and this part of the project will be done in collaboration with Andrea Malchiodi and Felix Schulze. The second part of the project concerns minimal graphs over disjoint domains of R m, that aims to answer a conjecture posed in [11]. The conjecture, arising from the study of properly embedded minimal surfaces, concerns the maximal number s(m) of disjoint domains in R m supporting nontrivial minimal graphs with zero boundary value. A sharp bound for s(m) is an open problem, and even its asymptotic behaviour as m is currently unknown. W. Meeks conjectured that s(2) = 2, and the problem is still open in full generality. This project, in collaboration with V. Tkachev, T. Bourni and G. Tinaglia, aims to prove Meeks conjecture and to find sharper ways to estimate s(m). Progress in this direction are expected to lead to new effective tools to study the geometry of proper minimal hypersurfaces. 1

2 State of the art and motivations Isoperimetric inequalities on Cartan-Hadamard manifolds The proofs of the Cartan-Hadamard conjecture in dimensions n = 2, 3, 4 that can be found, respectively, in [3], [9] and [5], rely on quite different methods: for n = 2, the theorem can be proved via integral identities involving the Jellett-Hsiung Minkovski formula, while for n = 3 the problem has been addressed via the inverse mean curvature flow. In dimension n = 4, on the other hand, the sharp constant was obtained via integral geometry on the unit tangent bundle of Ω. This last approach works in any dimension and gives an explicit S n, but only S 4 is sharp. Although the method seems likely to refine to yield sharp S n, to present no one succeeded to twist the argument and get the conjecture. Other effective methods to achieve the isoperimetric inequality on R n, that seem quite promising in a Cartan-Hadamard setting, use optimal transport (cf. [4, 2]) or the Alexandrov-Bakelman-Pucci inequality, ABP (cf. [1, 17]). However, both of them at some point crucially need the vector space structure of R n, and a way to overcome the problem seems to be challenging. In summary, although various promising approaches to tackle the conjecture are still available in the literature, no one adapt in a relatively easy way. It seems that new ideas are needed, and we believe that a solution of the Cartan-Hadamard conjecture would bring to us news tools to study manifolds with non-positive curvature. In particular, it may possibly increase our undestanding of how ABP and optimal transport methods could be effectively used on Cartan-Hadamard manifolds, and consequently open the way to the investigation of the isoperimetric conjecture for minimal submanifolds. Disjoint minimal graphs The problem of estimating the maximal number s(m) of disjoint domains in R m that support a nonconstant minimal graph with zero boundary value captured the attention of researchers since the influential paper [11], in connection with the study of the slices of properly embedded surfaces via hyperplanes. So far, we know by [10] (and refinement in [16]) that s(m) < e(m + 1) 2, but the approach, very ingenious, seems however difficult to improve so as to get a sharp upper bound. In the two dimensional case other tools are available, that were used by J. Spruck [14] to prove Meeks conjecture s(2) = 2 under suitable (binding) decay on u, 2 u (cf. also [18]). With no assumptions on u, the best known result is s(2) 3 in [16]. Our interest in solving Meeks conjecture and in obtaining better estimates for s(m) is primarily due to the strength of the techniques involved, relying on an appropriate use of density, frequency and projective volume functions to estimate the spreading of graphs at infinity. Their use is inspired by classical results in complex variable and harmonic function theories, but adaptation to the mean curvature operator appears to be quite challenging if one wishes to obtain sharp results with no assumptions on u. We feel that progress in the solution of the above problems would yield important new tools in minimal hypersurface theory. Depending on the specific results obtained, the project may open the road to a long-term investigation of proper minimal hypersurfaces. 2

3 Description and aims of the project Isoperimetric inequalities on Cartan Hadamard manifolds Both the method devised by C. Croke [5] and possible adaptations of the ones using optimal transport or ABP in [4, 1, 17] need a careful study of the geodesic flow Φ t on the unit tangent bundle π : SΩ Ω. The starting point of Croke s method, in particular, is Santalo s formula f(v)dv = f ( Φ t (v) ) ν π(v), v dσ(v) SΩ S + Ω for every smooth function f on SΩ. It is therefore important to understand how the geodesic flow behaves on SΩ. It is known that Φ t preserves the natural Liouville measure on SΩ, that is the volume measure associated to the Sasaki metric on SΩ. It seems that a more detailed study of the link between Φ t and the Sasaki metric be needed. The same problem appears when trying to adapt the ABP method in [1] and the one via optimal transport in [?, 2]. The method in [1] considers a solution of the Neumann problem { u = vol( Ω) vol(ω) on Ω R n u ν = 1 on Ω. and the isoperimetric inequality is deduced by studying the map u : Ω R n restricted to the contact set { Σ = x Ω : x is a minimum of u(y) p, y, for some p R n}. Although a suitable replacement for the Legendre transform is not difficult to find on a Cartan- Hadamard manifold, the geometry of the map u on the contact set is quite more complicated, the main problem being how to identify tangent spheres at different points of Ω: indeed, a natural identification is possible, but it is far from having the minimal regularity requirements to be used. The same hurdle appear when using optimal transport techniques. To present, it seems to us that this regularity problem be quite challenging, and we suspect that a brand new approach is needed. Apart from the conjecture itself, in view of the range of applications of ABP estimates and of optimal transport methods in Geometric Analysis a more thorough understanding of them on manifolds with non-positive curvature will be important, and is intended to be a long-term project. Ideas that could come from this study are expected to be useful to investigate the isoperimetric inequality for minimal submanifolds. Our purposes in this line of research would therefore be the following ones: Investigate and prove the Cartan-Hadamard conjecture in any dimension. Investigate how to get sharp ABP estimates, and how to use optimal transport techniques, on Cartan-Hadamard manifolds. Investigate the isoperimetric inequality for minimal submanifolds. Counting disjoint minimal graphs In the literature, the techniques to estimate s(m) differ considerably according to whether m = 2 or m 3. If m 3, currently the only effective known method is the one in [10], later refined in [16]. It is based on a careful use of the monotonicity of the mean value Θ u (r) = 1 V (r) 3 Σ B r u,

4 where Σ is the graph of u, V (t) = ω m r m and ω m is the volume of the unit sphere S m. However, it seems hard to adapt the ideas in [10] further to get a sharp bound for s(m), at least without additional conditions on u. Indeed, even identifying the right asymptotic behaviour of s(m) is an open problem, in view of the paucity of examples of minimal hypersurfaces in higher dimensions. On the other hand, if m = 2 the problem has been tackled by adapting tools from harmonic function theory and from classical Phragmén-Lindelöf theorems in complex analysis. For instance, in [14] the author proved s(2) = 2 provided that u 0 and 2 u decays quadratically as x. These assumptions are crucial to make the mean curvature operator close enough, for large x, to the Laplace-Beltrami operator in order to use the classical frequency function. However, the frequency seems to work well for the mean curvature operator only under the above assumptions on u, 2 u. To the best of our knowledge, a definition of frequency tailored to the mean curvature operator independently of the behaviour of u is currently unknown. A different approach that seems very promising at least in the 2-dimensional case, is the one developed in [15, 18] via the use of the projective volume Ψ(r) = 1 log V (r) Σ (B r\b 1) dx V ( x ). Exploiting Ψ, in [15] the bound s(2) 3 is obtained with no assumptions on u. The projective volume is a classical tool in the study of conformal maps of the plane, but it is not of common use in minimal hypersurface theory. Because of its properties, it seems promising that a refinement of the techniques in [16], together with new ideas possibily related to those in [10, 14], could lead to our first purpose: prove Meeks conjecture s(2) = 2 in full generality. However, up to now a crucial lemma limits the effectiveness of the approach in [14, 18, 16] to dimension m = 2. Therefore, in the higher-dimensional case it seems that brand new ideas are needed for our second aim: obtain a sharper estimate for s(m), possibly identifying the optimal growth of s(m) as m. Clearly, the identification of the order of growth of s(m) requires knowing examples of disjoint admissible domains, for instance studying the zero level sets of entire minimal graphs in dimension m 8. As for the original Bernstein problem, a careful investigation of minimal cones, in particular of the polynomial ones in [13], should be very helpful. References cabre castillon chavel CNV [1] X. Cabré, Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman- Pucci method: a survey. Chin. Ann. Math. Ser. B 38 (2017), no. 1, [2] P. Castillon, Submanifolds, isoperimetric inequalities and optimal transportation. J. Funct. Anal. 259 (2010), no. 1, [3] I. Chavel, Riemannian Geometry: A Modern Introduction. 2nd Revised Edition, Cambridge University Press, Cambridge, [4] C. Cordero-Erausquin, B. Nazaret and C. Villani, A mass transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math., 182, 2004, croke [5] C.B. Croke, A sharp four-dimensional isoperimetric inequality. Comment. Math. Helv. 59 (1984),

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