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Register a new userArea minimizers in a K3 surface and holomorphicity
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A well known consequence of the Wirtinger inequality is that in a Kaehler surface a holomorphic curve is an area minimizer in its homology class. In light of this result it is natural, given a Kaehler surface, to investigate the relation between area minimizers and complex curves. When the Kaehler surface is a K3 surface this problem takes on a new character. A Ricci flat (Calabi-Yau) metric on a K3 surface X is hyperkaehler in the sense that there is a two-sphere of complex structures, called the hyperkaehler line, each of which is compatible with the metric. A minimizer of area among surfaces representing a homology class alpha consists of a sum of branched immersed surfaces and it is then reasonable to ask whether each surface in this collection is holomorphic for some complex structure on the hyperkaehler line. Though this is true for many homology classes and there is other evidence that makes this pausible, in this paper we show that there is an integral homology class alpha and a hyperkaehler metric g such that no area minimizer of alpha has this property.
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Let $fcolon M^{2n}tomathbb{R}^{2n+p}$ denote an isometric immersion of a Kaehler manifold of complex dimension $ngeq 2$ into Euclidean space with codimension $p$. If $2pleq 2n-1$, we show that generic rank conditions on the second fundamental form of the submanifold imply that $f$ has to be a minimal submanifold. In fact, for codimension $pleq 11$ we prove that $f$ must be holomorphic with respect to some complex structure in the ambient space.
Let $Kbackslash G$ be an irreducible Hermitian symmetric space of noncompact type and $Gamma ,subset, G$ a closed torsionfree discrete subgroup. Let $X$ be a compact Kahler manifold and $rho, :, pi_1(X, x_0),longrightarrow, Gamma$ a homomorphism such that the adjoint action of $rho(pi_1(X, x_0))$ on $text{Lie}(G)$ is completely reducible. A theorem of Corlette associates to $rho$ a harmonic map $X, longrightarrow, Kbackslash G/Gamma$. We give a criterion for this harmonic map to be holomorphic. We also give a criterion for it to be anti--holomorphic.
The variational problem for the functional $F=frac12|phi^*omega|_{L^2}^2$ is considered, where $phi:(M,g)to (N,omega)$ maps a Riemannian manifold to a symplectic manifold. This functional arises in theoretical physics as the strong coupling limit of the Faddeev-Hopf energy, and may be regarded as a symplectic analogue of the Dirichlet energy familiar from harmonic map theory. The Hopf fibration $pi:S^3to S^2$ is known to be a locally stable critical point of $F$. It is proved here that $pi$ in fact minimizes $F$ in its homotopy class and this result is extended to the case where $S^3$ is given the metric of the Bergers sphere. It is proved that if $phi^*omega$ is coclosed then $phi$ is a critical point of $F$ and minimizes $F$ in its homotopy class. If $M$ is a compact Riemann surface, it is proved that every critical point of $F$ has $phi^*omega$ coclosed. A family of holomorphic homogeneous projections into Hermitian symmetric spaces is constructed and it is proved that these too minimize $F$ in their homotopy class.
The conformal parameterisation of a minimal surface is harmonic. Therefore, a minimal surface is a critical point of both the energy functional and the area functional. In this paper, we compare the Morse index of a minimal surface as a critical point of the area functional with its Morse index as a critical point of the energy functional. The difference between these indices is at most the real dimension of Teichmuller space. This comparison allows us to obtain surprisingly good upper bounds on the index of minimal surfaces of finite total curvature in Euclidean space of any dimension. We also bound the index of a minimal surface in an arbitrary Riemannian manifold by the area and genus of the surface, and the dimension and geometry of the ambient manifold.
We establish a quantitative lower bound on the reach of flat norm minimizers for boundaries in $mathbb{R}^2$.
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