作者: agoohuang

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Communications in Contemporary Mathematics
(2024) 2350064 (44 pages)
The Author(s)
DOI: 10.1142/S0219199723500645

Wu-Hsiung Huang
Department of Mathematics
National Taiwan University, Taipei 106, Taiwan
whuang0706@gmail.com

Received 15 September 2022
Revised 1 November 2023
Accepted 30 November 2023
Published 28 February 2024

In this paper, we establish a “global” Morse index theorem. Given a hypersurface Mⁿ of constant mean curvature, immersed in Rⁿ⁺¹. Consider a continuous deformation of “generalized” Lipschitz domain D(t) enlarging in Mⁿ. The topological type of D(t) is permitted to change along t, so that D(t) has an arbitrary shape which can “reach afar” in Mⁿ, i.e. cover any preassigned area. The proof of the global Morse index theorem is reduced to the continuity in t of the Sobolev space H_t of variation functions on D(t), as well as the continuity of eigenvalues of the stability operator. We devise a “detour” strategy by introducing a notion of “set-continuity” of D(t) in t to yield the required continuities of H_t and of eigenvalues. The global Morse index theorem thus follows and provides a structural theorem of the existence of Jacobi fields on domains in Mⁿ.

Keywords: Constant mean curvature; Sobolev space of variations; Jacobi field; Morse index theorem; continuity of eigenvalues.

Mathematics Subject Classification 2020: 53C23, 58E12, 35J25, 46E35, 49R05

全文檔案： huang-2024-a-global-morse-index-theorem-and-applications-to-jacobi-fields-on-cmc-surfaces