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Additional resources for Donaldson, Differential Geometry - Geometry of Four Manifolds
Superficially, this situation resembles the problem of Ginzburg-Landau vortices studied by Bethuel et al. [BBH94] and others. ) In this case, however, the analogy is not good and we need completely different tools. For this choice of the potential function, we expect needle-shaped surfaces. We analyse surfaces of revolution in particular and we determine a limiting energy for this special case. Furthermore, we have some estimates for more general surfaces. In contrast to the first problem, it is not clear how to generalise the theory to similar potential functions.
Step 2. The second derivatives of r may be bounded in terms of a bound for the mean curvature H of the surface r as follows. The expression of H implies g11 H = 1 1 2 2 − 2h 12 g11 g12 + h 11 g12 ) + h 11 . (h 22 g11 2|g| 2 Since the surface is convex and the unit normal n was chosen to be an inner normal, the quadratic form on the right-hand side is positive definite so that we have h 11 < 2g11 H. Similarly, we have h 22 < 2g22 H. Then we get √ |h 12 | < 2H g11 g22 , since (h ij ) is positive definite.
He also pointed out that a reasonable sufficient condition might be the decay rate of the Gauss curvature at infinity. In 1993, Hong [Hon93] gave an affirmative answer and showed that a correct sufficient condition is that the Gauss curvature decays at infinity faster than the inverse square of the geodesic distance. 4 Let (M, g) be a complete simply connected smooth surface with Gauss curvature K < 0 and (ρ, θ) be a (global) geodesic polar coordinate. Assume, for some constant δ > 0, (H1 ) (H2 ) ρ2+δ |K | is decreasing in ρ outside a compact set; ∂θi ln |K |, (i = 1, 2), ρ∂ρ ∂θ ln |K | are bounded.
Donaldson, Differential Geometry - Geometry of Four Manifolds