It has been shown in by Huang-Lucia-Tarantello [17] that, for given $\vert c \vert <1$, the moduli space of constant mean curvature (CMC) $c$-immersions of a closed orientable surface of genus $\mathfrak{g} \geq 2$ into a hyperbolic $3$-manifold can be parametrized by elements of the tangent bundle of the corresponding Teichmüller space. This is attained by showing the unique solvability of the Gauss-Codazzi equations governing (CMC) c-immersions. The corresponding unique solution is identified as the global minimum (and only critical point) of the Donaldson functional $D_t$ (introduced in [11]) given in (1.3) with $t=1-c^{2}$.
When $\vert c \vert \geq 1$ (i.e. $t\leq 0$), so far nothing is known about the existence of analogous (CMC) c-immersions. Indeed, for $t\leq 0$ the functional $D_{t}$ may no longer be bounded from below and evident non-existence situations do occur.
Already the case $\vert c \vert =1$ (i.e. $t=0$) appears rather involved and actually (CMC) 1-immersions can be attained only as "limits" of (CMC) c-immersions for $\vert c \vert \longrightarrow 1^{-}$. To handle this situation, here we analyse the asymptotic behaviour of minimizers of $D_{t}$ as $t \longrightarrow 0^{+}$.
We use an accurate asymptotic analysis to describe possible blow-up phenomena. In this way, we can relate the existence of (CMC) 1-immersions to the Kodaira map. As a consequence, we obtain the first existence and uniqueness result about (CMC) 1-immersions of surfaces of genus $\mathfrak{g}=2$ into hyperbolic 3-manifolds.
In this paper we show a quantitative rigidity result for the minimizer of the Willmore functional among all projective planes in $\mathbb{R}^n$ with $n\ge 4$. We also construct an explicit counterexample to a corresponding rigidity result in codimension one, by showing that an Enneper surface might split-off during a blow-up process. For conformal immersions of spheres with large enough Willmore energies, we construct explicit counterexamples to a quantitative rigidity result and this complements the recently obtained rigidity results in [LaNg13].
In this paper we define a Donaldson type functional whose Euler-Lagrange equations are a system of differential equations which corresponds to Hitchin's self-duality equations for a suitable choice of Higgs bundle on closed Riemann surfaces. The main challenge of this functional is its lack of regularity and lack of compactness when defined in its natural domain of definition. Though a standard variational approach cannot directly be applied, we provide the appropriate analytical tools that make Donaldson functional treatable by a variational viewpoint. We prove that this functional admits a u…
We consider the problem of minimizing the Willmore energy in the class of conformal immersions of a given closed, genus p Riemann surface into R^n for n=3,4. We prove existence of a smooth minimizer, provided that the infimum is below a certain bound ${\cal W}(n,p)$. For tori in R^3 we have explicitely ${\cal W}(3,1) = 8π$.
Given a Riemannian 3-ball $(\bar B, g)$ of non-negative scalar curvature, Bartnik conjectured that $(\bar B, g)$ admits an asymptotically flat (AF) extension (without horizons) of the least possible ADM mass, and that such a mass-minimizer is an AF solution to the static vacuum Einstein equations, uniquely determined by natural geometric conditions on the boundary data of $(\bar B, g)$. We prove the validity of the second statement, i.e.~such mass-minimizers, if they exist, are indeed AF solutions of the static vacuum equations. On the other hand, we prove that the first statement is not true …
The paper is devoted to the variational analysis of the Willmore, and other L^2 curvature functionals, among immersions of 2-dimensional surfaces into a compact riemannian m-manifold (M^m,h) with m>2. The goal of the paper is twofold, on one hand, we give the right setting for doing the calculus of variations (including min max methods) of such functionals for immersions into manifolds and, on the other hand, we prove existence results for possibly branched Willmore spheres under various constraints (prescribed homotopy class, prescribed area) or under curvature assumptions for M^m. To this ai…
We show that the well-known family of $2$-lobed Delaunay tori $\;f^b\;$ in $\;S^3,\;$ parametrized by $\;b \in \mathbb R_{\geq1},\;$ uniquely minimizes the Willmore energy among all immersions from tori into $3$-space of conformal class $\;(a, b)\;$. As a corollary we obtain an alternate proof of the Willmore conjecture in $3$-space. This new strategy can be generalized to arbitrary codimensions provided a classification of isothermic constrained Willmore tori is possible and all $\;f^b\;$ remain stable in all codimensions.
Let $X$ be a compact Kähler manifold with a given ample line bundle $L$. In \cite{Don05}, Donaldson proved that the Calabi energy of a Kähler metric in $c_1(L)$ is bounded from below by the supremum of a normalized version of the minus Donaldson--Futaki invariants of test configurations of $(X,L)$. He also conjectured that the bound is sharp. In this paper, we prove a metric analogue of Donaldson's conjecture, we show that if we enlarge the space of test configurations to the space of geodesic rays in $\mathcal{E}^2$ and replace the Donaldson--Futaki invariant by the radial Mabuchi K-energy $\…
We investigate isometric immersions of disks with constant negative curvature into $\mathbb{R}^3$, and the minimizers for the bending energy, i.e. the $L^2$ norm of the principal curvatures over the class of $W^{2,2}$ isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in $\mathbb{H}^2$ into $\mathbb{R}^3$. In elucidating the connection between these immersions and the non-existence/singularity results of Hilbert and Amsler, we obtain a lower bound for the $L^\infty$ norm of the principal curvatures for such smooth isometric immersions. We also …
Extending Donaldson's HyperKähler Structure via Minimal Surfaces into $\mathrm{PSL}(2,\mathbb{C})$
University of Turin（都灵大学）；University of California Riverside（加州大学河滨分校）；Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)（数学科学研究所（西班牙高等科研理事会-马德里自治大学-马德里理工大学-马德里 Complutense 大学））；University of Pisa（比萨大学）
We consider smooth isotropic immersions from the 2-dimensional torus into $R^{2n}$, for $n \geq 2$. When $n = 2$ the image of such map is an immersed Lagrangian torus of $R^4$. We prove that such isotropic immersions can be approximated by arbitrarily $C^0$-close piecewise linear isotropic maps. If $n \geq 3$ the piecewise linear isotropic maps can be chosen so that they are piecewise linear isotropic immersions as well. The proofs are obtained using analogies with an infinite dimensional moment map geometry due to Donaldson. As a byproduct of these considerations, we introduce a numerical flo…
In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the $K$-energy is non-increasing. Moreover, we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of Kähler potentials implies th…
We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with uniformly bounded areas and Willmore energies. The compactness property is applied to construct Willmore type surfaces in compact Riemannian manifolds. This includes (a) existence of a Willmore 2-sphere in ${\mathbb S}^n$ with at least 2 nonremovable singular points (b) existence of minimizers of the Willmore functional with prescribed area in a compact mani…
Let (M,ω) be a symplectic manifold, and (Σ,σ) a closed connected symplectic 2-manifold. We construct a weakly symplectic form {ω^{D}}_{(Σ, σ)} on the space of immersions Σ\to M that is a special case of Donaldson's form. We show that the restriction of {ω^{D}}_{(Σ,σ)} to any orbit of the group of Hamiltonian symplectomorphisms through a symplectic embedding (Σ,σ) \to (M,ω) descends to a weakly symplectic form ω^D_{\red} on the quotient by Sympl(Σ,σ), and that the obtained symplectic space is a symplectic quotient of the subspace of symplectic embeddings S_{e}(Σ,σ) with respect to the Sympl(Σ,σ…
We prove that the critical points of various energies such as the area, the Willmore energy, the frame energy for tori...etc among possibly branched immersions constrained to evolve within a smooth sub-manifold of the Teichmüller space satisfy the corresponding constrained Euler Lagrange equation. We deduce that critical points of the Willmore energy or the frame energy for tori are smooth analytic surfaces, away possibly from isolated branched points, under the condition that either the genus is at most 2 or if the sub-manifold does not intersect the subspace of hyper-elliptic points. Using a…
We can define the ``volume'' $V$ for Hermitian connections on a Hermitian complex line bundle over a Riemannian manifold $X$, which can be considered to be the ``mirror'' of the standard volume for submanifolds. This is called the Dirac-Born-Infeld (DBI) action in physics. In this paper, (1) we introduce the negative gradient flow of $V$, which we call the line bundle mean curvature flow. Then, we show the short-time existence and uniqueness of this flow. When $X$ is Kähler, we relate the negative gradient of $V$ to the angle function and deduce the mean curvature for Hermitian metrics on a ho…
We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf. We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin-Tholozan, we prove that any representation from a finitely generated free group to the isomet…
We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold $(M,h)$ without boundary. First, under the assumption that $(M,h)$ is the euclidean 3-space endowed with a semi-perturbed metric with perturbation small in $C^1$ norm and of compact support, we prove that if there is some point $\bar{x} \in M$ with scalar curvature $R^M(\bar{x})>0$ then there exists a smooth embedding $f:S^2 \hookrightarrow M$ minimizing the Willmore functional $1/4\int |H|^2$, where $H$ is the mean curvature. Second, assuming that $(M,h)$ is of bounded geometry (i.e. bo…
