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Register a new userThe Donaldson equation
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Authors
Weiyong He
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In this short note, we solve a Dirichlet problem for a fully nonlinear elliptic equation. The operator is introduced by S. Donaldson and it is relevant to the geometry of the space of volume forms.
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This article introduces the degenerate special Lagrangian equation (DSL) and develops the basic analytic tools to construct and study its solutions. The DSL governs geodesics in the space of positive graph Lagrangians in $mathbb{C}^n.$ Existence of geodesics in the space of positive Lagrangians is an important step in a program for proving existence and uniqueness of special Lagrangians. Moreover, it would imply certain cases of the strong Arnold conjecture from Hamiltonian dynamics. We show the DSL is degenerate elliptic. We introduce a space-time Lagrangian angle for one-parameter families of graph Lagrangians, and construct its regularized lift. The superlevel sets of the regularized lift define subequations for the DSL in the sense of Harvey--Lawson. We extend the existence theory of Harvey--Lawson for subequations to the setting of domains with corners, and thus obtain solutions to the Dirichlet problem for the DSL in all branches. Moreover, we introduce the calibration measure, which plays a r^ole similar to that of the Monge--Amp`ere measure in convex and complex geometry. The existence of this measure and regularity estimates allow us to prove that the solutions we obtain in the outer branches of the DSL have a well-defined length in the space of positive Lagrangians.
The Special Lagrangian Potential Equation for a function $u$ on a domain $Omegasubset {bf R}^n$ is given by ${rm tr}{arctan(D^2 ,u) } = theta$ for a contant $theta in (-n {piover 2}, n {piover 2})$. For $C^2$ solutions the graph of $Du$ in $Omegatimes {bf R}^n$ is a special Lagrangian submanfold. Much has been understood about the Dirichlet problem for this equation, but the existence result relies on explicitly computing the associated boundary conditions (or, otherwise said, computing the pseudo-convexity for the associated potential theory). This is done in this paper, and the answer is interesting. The result carries over to many related equations -- for example, those obtained by taking $sum_k arctan, lambda_k^{{mathfrak g}} = theta$ where ${{mathfrak g}} : {rm Sym}^2({bf R}^n)to {bf R}$ is a Garding-Dirichlet polynomial which is hyperbolic with respect to the identity. A particular example of this is the deformed Hermitian-Yang-Mills equation which appears in mirror symmetry. Another example is $sum_j arctan kappa_j = theta$ where $kappa_1, ... , kappa_n$ are the principal curvatures of the graph of $u$ in $Omegatimes {bf R}$. We also discuss the inhomogeneous Dirichlet Problem ${rm tr}{arctan(D^2_x ,u)} = psi(x)$ where $psi : overline{Omega}to (-n {piover 2}, n {piover 2})$. This equation has the feature that the pull-back of $psi$ to the Lagrangian submanifold $Lequiv {rm graph}(Du)$ is the phase function $theta$ of the tangent spaces of $L$. On $L$ it satisfies the equation $ abla psi = -JH$ where $H$ is the mean curvature vector field of $L$.
In this paper, we solve the Dirichlet problem with continuous boundary data for the Lagrangian mean curvature equation on a uniformly convex, bounded domain in $mathbb{R}^n$.
We study the wave equation on infinite graphs. On one hand, in contrast to the wave equation on manifolds, we construct an example for the non-uniqueness for the Cauchy problem of the wave equation on graphs. On the other hand, we obtain a sharp uniqueness class for the solutions of the wave equation. The result follows from the time analyticity of the solutions to the wave equation in the uniqueness class.
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