Existence and nonexistence of patterns on Riemannian manifolds

Abstract

We study existence and nonexistence of patterns on Riemannian manifolds, depending on the Ricci curvature of the manifold and suitable assumptions on the boundary. In the case of surfaces of revolutions in R(3), necessary and sufficient conditions for existence of patterns are given. (C) 2011 Elsevier Inc. All rights reserved

The existence of patterns on surfaces of revolution without boundary

“…In this paper we shall extend the existence result for patterns given in [4] to the case of a surface of revolution without boundary. In considering this case, we are motivated by interesting applications in biology (e.g., see [10,11]).…”

“…The existence and nonexistence of patterns on connected compact n-dimensional Riemannian manifolds have been studied in [4][5][6][7][8]. Note that the case ∂M ̸ = ∅ has also been considered; in that situation problem (1.1) is completed with homogeneous zero Neumann boundary conditions.…”

“…In particular, in [6] (see [4] for a detailed proof) it is stated that no pattern exists if the following assumptions are satisfied:…”

“…Furthermore, in the special case where the Riemannian manifold is a surface of revolution of R 3 (with or without boundary), the previous nonexistence result has been generalized in [4] by adapting some ideas of [9]. To be specific, without making assumptions on the second fundamental form of the boundary ∂M, it is shown that for problem (1.1) no pattern exists when the sum of the Ricci curvature at every point p ∈ M and the square of the geodesic curvature of the parallel passing through p is nonnegative (see Remark 2.1(i)).…”

“…In considering this case, we are motivated by interesting applications in biology (e.g., see [10,11]). In general, we follow the same line of arguments as [4]; however, to do this we overcome several technical difficulties arising because the boundary is empty.…”