Minimal surfaces and deformations of holomorphic curves in Kähler-Einstein manifolds

Claudio Arezzo

Displaying similar documents to “Minimal surfaces and deformations of holomorphic curves in Kähler-Einstein manifolds”

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In this paper we give a survey of methods of Quaternionic Holomorphic Geometry and of applications of the theory to minimal surfaces. We discuss recent developments in minimal surface theory using integrable systems. In particular, we give the Lopez–Ros deformation and the simple factor dressing in terms of the Gauss map and the Hopf differential of the minimal surface. We illustrate the results for well–known examples of minimal surfaces, namely the Riemann minimal surfaces and the...

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We show that the apparatus of support functions, usually used in convex surfaces theory, leads to the linear equation Δh + 2h = 0 describing locally germs of minimal surfaces. Here Δ is the Laplace-Beltrami operator on the standard two-dimensional sphere. It explains the existence of the sum operator of minimal surfaces, introduced recently. In 4-dimensional space the equation Δ h + 2h = 0 becomes inequality wherever the Gauss curvature of a minimal hypersurface is nonzero.

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Quaternionic maps and minimal surfaces

Jingyi Chen, Jiayu Li (2005)

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Let $(M, J^{α}, α = 1, 2, 3)$ and $(N, 𝒥^{α}, α = 1, 2, 3)$ be hyperkähler manifolds. We study stationary quaternionic maps between $M$ and $N$ . We first show that if there are no holomorphic 2-spheres in the target then any sequence of stationary quaternionic maps with bounded energy subconverges to a stationary quaternionic map strongly in $W^{1, 2} (M, N)$ . We then find that certain tangent maps of quaternionic maps give rise to an interesting minimal 2-sphere. At last we construct a stationary quaternionic map with a codimension-3 singular set by...