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Stationary Critical Points of the Heat Flow in Spaces of Constant Curvature
Author(s) -
Sakaguchi Shigeru
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1017/s0024610700001976
Subject(s) - mathematics , mathematical analysis , heat equation , bounded function , euclidean space , initial value problem , geodesic , critical point (mathematics) , domain (mathematical analysis) , flow (mathematics) , geometry
The paper considers stationary critical points of the heat flow in sphere S N and in hyperbolic space H N , and proves several results corresponding to those in Euclidean space R N which have been proved by Magnanini and Sakaguchi. To be precise, it is shown that a solution u of the heat equation has a stationary critical point, if and only if u satisfies some balance law with respect to the point for any time. In Cauchy problems for the heat equation, it is shown that the solution u has a stationary critical point if and only if the initial data satisfies the balance law with respect to the point. Furthermore, one point, say x 0 , is fixed and initial‐boundary value problems are considered for the heat equation on bounded domains containing x 0 . It is shown that for any initial data satisfying the balance law with respect to x 0 (or being centrosymmetric with respect to x 0 ) the corresponding solution always has x 0 as a stationary critical point, if and only if the domain is a geodesic ball centred at x 0 (or is centrosymmetric with respect to x 0 , respectively).