Nonmaximality of known extremal metrics on torus and Klein bottle. (English. Russian original) Zbl 1290.58008

Karpukhin, M. A.

Sb. Math. 204, No. 12, 1728-1744 (2013); translation from Mat. Sb. 204, No. 12, 31-48 (2013).

Let \(\Delta\) be the Laplacian on a compact connected Riemann surface \((M,g)\). Let \[ \{0=\lambda_0<\lambda_1(M,g)\leq\lambda_2(M,g)\dots\} \] be the spectrum of \((M,g)\) where each eigenvalue is repeated according to multiplicity. To take into account rescaling, let \(\Lambda_k(M,g):=\lambda_k(M,g)\cdot\text{vol}(M,g)\). Let \(g_t\) be an analytic deformation of \(g\). One says \((M,g)\) is \(\Lambda_i\) extremal if \(\partial_t\Lambda_i(M,g_t)|_{t=0}=0\). The metric is called maximal if the value \(\Lambda_i(M,g)\) is maximal amongst all metrics \(g\) on \(M\). The following are the known extremal metrics:

1.

The Otsuki Torus \(O_{p/q}\) (see [A. V. Penskoi, Math. Nachr. 286, No. 4, 379–391 (2013; Zbl 1271.58007)]).

2.

The Lawson torus and the Klein bottle \(\tau_{m,k}\) (see [A. V. Penskoi, Mosc. Math. J. 12, No. 1, 173–192 (2012; Zbl 1272.58010)])

3.

Metrics on the surfaces \(\tilde\tau_{m,k}\) bipolar to the Lawson surface (see [H. Lapointe, Differ. Geom. Appl. 26, No. 1, 9–22 (2008; Zbl 1139.58025)])

4.

Metrics on the bipolar surfaces to Otsuki tori \(\tilde O_{p/q}\). (see [M. A. Karpukhin, “Spectral properties of bipolar surfaces to Otsuki tori”, arXiv:1205.6316]).

The main result of the paper is to establish that there are no maximal metrics amongst the metrics given above except for \(\tilde\tau_{3,1}\). The author also shows the metric on the Clifford torus is extremal for an infinite number of the functionals \(\Lambda_i\) but is not maximal for any of them.

Reviewer: Peter B. Gilkey (Eugene)

MSC:

58E11

Critical metrics

Keywords:

extremal metrics; bipolar surface; Otsuki tori; Lawson tau surface

Citations:

Zbl 1271.58007; Zbl 1272.58010; Zbl 1139.58025