A gap theorem for Willmore tori and an application to the Willmore flow. (English) Zbl 1286.53012
Summary: In 1965, Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in \(\mathbb R^3\) is at least \(2\pi^2\) and attains this minimal value if and only if the torus is a Möbius transform of the Clifford torus. This was recently proved by F. C. Marques and A. Neves [Ann. Math. (2) 179, No. 2, 683–782 (2014; Zbl 1297.49079)]. In this paper, we show for tori that there is a gap to the next critical point of the Willmore energy and we discuss an application to the Willmore flow. We also prove an energy gap from the Clifford torus to surfaces of higher genus.
MSC:
| 53A30 | Conformal differential geometry (MSC2010) |
Keywords:
Willmore functional; surface theory; nonlinear elliptic equation of 4th order; geometric flowsCitations:
Zbl 1297.49079References:
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