Upper bound of Kähler angles on the β-symplectic critical surfaces

Abstract

Let (M,g) be a Kähler surface and \mathrm{\Sigma } be a \beta-symplectic critical surface in M. If L_q\left(\mathrm{\Sigma }\right) is bounded for some q>3, then we give a uniform upper bound for the Kähler angle on \mathrm{\Sigma }. This bound only depends on M,q,\beta and the L_q functional of \mathrm{\Sigma }. For q>4, this estimate is known and we extend the scope of q.

Keywords

Kähler surface / β-symplectic critical surfaces / Kähler angle / L_β functional

References

[1] Arezzo C. Minimal surfaces and deformations of holomorphic curves in Kähler-Einstein manifolds. Ann Scuola Norm Sup Pisa Cl Sci 2000; 29(2): 473–481

[2] Chern S S, Wolfson J. Minimal surfaces by moving frames. Amer J Math 1983; 105(1): 59–83

[3] Han X L, Li J Y. Symplectic critical surfaces in Kähler surfaces. J Eur Math Soc 2010; 12(2): 505–527

[4] Han X L, Li J Y. The second variation of the functional L of symplectic criticalsurfaces in Kähler surfaces. Commun Math Stat 2014; 2(3−4): 311–330

[5] Han X L, Li J Y, Sun J. The deformation of symplectic critical surfaces in a Kählersurface-Ⅰ. Int Math Res Not IMRN 2018; 20: 6290–6328

[6] Han X L, Li J Y, Sun J. The deformation of symplectic critical surfaces in a Kählersurface-Ⅱ– compactness. Calc Var Partial Differential Equations 2017; 56(3): 1–22

[7] Michael J H, Simon L M. Sobolev and mean-value inequalities on generalized submanifolds of Rⁿ. Comm Pure Appl Math 1973; 26: 361–379

[8] Micallef M, Wolfson J. The second variation of area of minimal surfaces in fourmanifolds. Math Ann 1993; 295(2): 245–267

[9] Webster S M. Minimal surfaces in a Kähler surface. J Differential Geom 1984; 20(2): 463–470

[10] Wolfson J. Minimal surfaces in Kähler surfaces and Ricci curvature. J Differential Geom 1989; 29(2): 281–294

[11] Wolfson J. On minimal surfaces in a Kähler manifold of constant holomorphic sectional curvature. Trans Amer Math Soc 1985; 290(2): 627–646

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Higher Education Press