Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds

Abstract

Our work proposes a unified approach to three different topics in a general Riemannian setting: splitting theorems, symmetry results and overdetermined elliptic problems. By the existence of a stable solution to the semilinear equation −∆u = f (u) on a Riemannian manifold with non-negative Ricci curvature, we are able to classify both the solution and the manifold. We also discuss the classification of monotone (with respect to the direction of some Killing vector field) solutions, in the spirit of a conjecture of De Giorgi, and the rigidity features for overdetermined elliptic problems on submanifolds with boundary. R ) = o(R 2 log R) as R → +∞. Remark 6. For R 2 , the above theorem generalizes the one-dimensional symmetry result in Theorem 1.2 of [18]. See also [37] for interesting studies on the geometric and topological properties of overdetermined problems in the Euclidean plane.By the monotonicity Theorem 1.1 in [4], the relation ∇u, X > 0 on Ω is authomatic for globally Lipschitz epigraphs Ω of Euclidean space and for some large class of nonlinearities f including the prototype Allen-Cahn one f (u) = u − u 3 (even without requiring the Neumann condition in (6)). However, it is an open problem to enlarge the class of domains Ω ⊆ R m for which ∇u, X > 0 is met, or to find nontrivial analogues on Riemannian manifolds. In the last section, we move some steps towards this problem by proving some lemmata that may have independent interest. In particular, we obtain the next result:Proposition 7. Let (M, , ) be a complete Riemannian manifold satisfying Ric ≥ −(m − 1)H 2 , , for some H ≥ 0, and let f ∈ C 1 (R) have the propertiesfor some λ > 0 and some small δ 0 , s 0 > 0. Let Ω ⊆ M be an open, connected subset, and suppose that u ∈ C 2 (Ω) ∩ C 0 (Ω) is a bounded, non-negative solution of −∆u = f (u) on Ω,

Monotonicity formulas in potential theory

“…and recalling the little discussion after formula (1.7), we obtain a clear geometric interpretation of the constancy of the P -function x → P (x) = |Du| 2 /u 2(n−1)/(n−2) (x), which is naturally associated with problem (1.1). The second step of our strategy consists in proving via a splitting principle that the metric g has indeed a product structure, provided the hypothesis of the Rigidity statement is satisfied (splitting techniques have been successfully employed in the context of partial differential equations for example in [3,23]). More in general, we use the above conformal reformulation of the original system combined with the Bochner identity to deduce the equation…”

Characterization of f-extremal disks

“…P. Sicbaldi [36] gave a counterexample of the BCN conjecture when n ≥ 3. Nevertheless, the BCN conjecture motivated interesting works as, for example, those of Farina and collaborators ( [15,16,17,18] and references therein). Recently, important contributions have been made in dimension n = 2.…”

Geometric rigidity of constant heat flow

“…The study of overdetermined problems is a very active and interesting field of research, lying at the border between geometry and analysis; for an overview, see for example [36], and then [7], [10], [11], [18], [19], [20], [23], [33], [34], [37], [41], [42], [43]; for problems in Riemannian manifolds see for example [12] and [13]. We stress that we assume compactness of Ω in this paper.…”