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For surfaces immersed into a compact Riemannian manifold, we consider the curvature functional given by the $L^{2}$ integral of the second fundamental form. We discuss an area bound in terms of the energy, with application to the existence of minimizers. This is joint work with V. Bangert.

53C44 ; 53C45

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An $R$-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each $R$-space has the canonical embedding into a Kähler $C$-space as a real form which is a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is an invariant under Hamiltonian isotopies and very fundamental to the study of the Floer homology for intersections of Lagrangian submanifolds. In this talk we provide a Lie theoretic formula for the minimal Maslov number of $R$-spaces canonically embedded in Einstein-Kähler $C$-spaces and discuss several examples of the calculation by the formula.
An $R$-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each $R$-space has the canonical embedding into a Kähler $C$-space as a real form which is a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is an invariant under Hamiltonian isotopies and very fundamental to the ...

53C55 ; 53C25

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We discuss singularities of Teichmüller harmonic map flow, which is a geometric flow that changes maps from surfaces into branched minimal immersions, and explain in particular how winding singularities of the map component can lead to singular behaviour of the metric component.

53C44 ; 30F60

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A minimal surface $M$ in the round sphere $\mathbb{S}^{n}$ is critical for area, as well as for the Willmore bending energy $W=\int\int(1+H^{2})da$. Willmore stability of $M$ is equivalent to a gap between −2 and 0 in its area-Jacobi operator spectrum. We show the $W$-stability of $M$ persists in all higher dimensional spheres if and only if the Laplacian of $M$ has first eigenvalue 2. The square Clifford 2-torus in $\mathbb{S}^{3}$ and the equilateral minimal 2-torus in $\mathbb{S}^{5}$ have this spectral gap, and each is embedded by first eigenfunctions, so both are "persistently” $W$-stable. On the other hand, we discovered the equilateral torus has nontrivial third variation (with vanishing second variation) of $W$, and thus is not a $W$-minimizer (though it is the $W$-minimizer if we fix the conformal type!). This is evidence the Willmore Conjecture holds in every codimension. Another result concerns higher genus minimal surfaces (such as those constructed by Lawson and those by Karcher-Pinkall-Sterling) in $\mathbb{S}^{3}$ which Choe-Soret showed are embedded by first eigenfunctions: we show their first eigenspaces are always 4-dimensional, and that this implies each is (up to Möbius transformations of $\mathbb{S}^{n}$) the unique $W$-minimizer in its conformal class. (Some analogous results hold for free boundary minimal surfaces in the unit ball $\mathbb{B}^{n}$....). This is joint work with Peng Wang.
A minimal surface $M$ in the round sphere $\mathbb{S}^{n}$ is critical for area, as well as for the Willmore bending energy $W=\int\int(1+H^{2})da$. Willmore stability of $M$ is equivalent to a gap between −2 and 0 in its area-Jacobi operator spectrum. We show the $W$-stability of $M$ persists in all higher dimensional spheres if and only if the Laplacian of $M$ has first eigenvalue 2. The square Clifford 2-torus in $\mathbb{S}^{3}$ and the ...

53C42

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