Pseudoconvexity for the special Lagrangian potential equation

Abstract

The Special Lagrangian Potential Equation for a function u on a domain $\Omega \subset {{\mathbb {R}}}^n$ is given by ${\mathrm{tr}}\{\arctan (D^2 \,u) \} = \theta$ for a contant $\theta \in (-n {\pi \over 2}, n {\pi \over 2})$. For $C^2$ solutions the graph of Du in $\Omega \times {{\mathbb {R}}}^n$ 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 ${{\mathfrak {g}}}: {\mathrm{Sym}}^2({{\mathbb {R}}}^n)\rightarrow {{\mathbb {R}}}$ is a Gårding-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 $\kappa _1,\ldots , \kappa _n$ are the principal curvatures of the graph of u in $\Omega \times {{\mathbb {R}}}$. We also discuss the inhomogeneous Dirichlet Problem

$$\begin{aligned} {\mathrm{tr}}\{\arctan (D^2_x \,u)\} = \psi (x) \end{aligned}$$

where $\psi : {\overline{\Omega }}\rightarrow (-n {\pi \over 2}, n {\pi \over 2})$. This equation has the feature that the pull-back of $\psi$ to the Lagrangian submanifold $L\equiv {\mathrm{graph}}(Du)$ is the phase function $\theta$ of the tangent spaces of L. On L it satisfies the equation $\nabla \psi = -JH$ where H is the mean curvature vector field of L.

Appendices

Appendix A. A Geometric interpretation of the inhomogeneous DP

The Eq. (A.1) below appeared as equation (2.18) in [21]. We left the proof as a exercise for the reader. However, this equation has an immediate consequence for the Dirichlet problem for the inhomogeneous SL equation (A.2), which is discussed in Sect. 5. This is given in Corollary A.2. It may have gone unnoticed and seems not to be well understood. For the convenience of the reader we give the proof of equation (2.18) in [21] here.

Proposition A.1

Let X be a Calabi-Yau manifold of complex dimension n. Let $\Phi$ be the parallel (n, 0)-form normalized so that ${\mathrm{Re}} \Phi$ has comass 1. Given $L\subset X$ an oriented Lagrangian submanifold, define the phase $\theta$ mod $2\pi$ by

$$\begin{aligned} \Phi \bigr |_L = e^{i\theta } d{\mathrm{vol}}_L. \end{aligned}$$

Then for any tangent vector field V on L, we have

$$\begin{aligned} V \theta = \langle JV, H\rangle , \end{aligned}$$

that is

$$\begin{aligned} \nabla \theta = -JH \end{aligned}$$

(A.1)

where H is the mean-curvature vector field of L, and J is the complex structure on X.

Proposition A.1 has the following immediate implication for the inhomogeneous SL potential equation ${\mathrm{tr}}\left\{ \arctan (D^2_x u) \right\} = \psi (x)$. Let $z\equiv x+iy \in {{\mathbb {R}}}^n\oplus i {{\mathbb {R}}}^n= {{\mathbb {C}}}^n$

Corollary A.2

Suppose $L\equiv \{(x, \nabla u(x)) : x\in \Omega \}$ is the graph of the gradient of $u\in C^2(\Omega )$ over a domain $\Omega \subset {{\mathbb {R}}}^n$. Then the inhomogeneous term

$$\begin{aligned} \theta (x) \ \equiv \ {\mathrm{tr}}\left\{ \arctan D_x^2 u \right\} , \end{aligned}$$

(A.2)

considered as a function on L, is the phase function for L. Thus it has gradient related to the mean curvature vector field H of L by

$$\begin{aligned} \nabla \theta = -JH \qquad {\mathrm{on}}\ \ L. \end{aligned}$$

(A.3)

Otherwise said, if u is a solution to the equation

$$\begin{aligned} {\mathrm{tr}}\left\{ \arctan (D^2_x u) \right\} = \psi (x) \end{aligned}$$

on $\Omega$, with $\psi (x)$ smooth, then

$$\begin{aligned} \nabla \widetilde{\psi }= -J H \qquad {\mathrm{on}}\ \ \ L \end{aligned}$$

(A.3)

where $\widetilde{\psi }$ is the pull-back of $\psi$ to L.

Note. Proposition A.1 is actually independent of the orientation of L. A change of orientation changes the function $\theta$ to $\theta +\pi$, and the conclusion is the same. In Corollary A.2, L is given the orientation of $\Omega$.

Proof of Proposition A.1

By a complex linear change of coordinates we may assume at p we have $\Phi = dz_1\wedge \cdots \wedge dz_n$. Now let $p=(x_0, \nabla u(x_0))$. The map $D^2 u$ is symmetric, so by a change of variables $(x,y) \rightarrow (gx, gy)$ for some $g\in {\mathrm{SO}}(n)$, we can assume that at $x_0$, $D^2 u$ is diagonal, i.e., $(D^2_{x_0} u)(\epsilon _k) = \lambda _k \epsilon _k$ for an orthonormal basis $\epsilon _1,\ldots , \epsilon _n$ of ${{\mathbb {R}}}^n$.

Now let $e_1,\ldots , e_n$ be an oriented orthonormal frame field on L in a neighborhood of $p=(x_0, \nabla u(x_0))$. Then

$$\begin{aligned} \Phi (e_1\wedge \cdots \wedge e_n) = e^{i\theta }, \end{aligned}$$

and so

$$\begin{aligned} V e^{i\theta } = e^{i\theta } i V\theta = \Phi \left( \sum _{k=1}^n e_1\wedge \cdots \wedge (\nabla _V e_k) \wedge \cdots \wedge e_n \right) \end{aligned}$$

since $\Phi$ is parallel. We may assume $\nabla ^L_{e_j} e_k = (\nabla _{e_j} e_k)^{{\mathrm{tang}}} = 0$ at $x_0$, so $\nabla _V e_k = ( \nabla _V e_k)^{{\mathrm{normal}}} = B_{V, e_k}$ the second fundamental form of L at $x_0$. Therefore we have

$$\begin{aligned} V e^{i\theta } = e^{i\theta } i V\theta = \Phi \left( \sum _{k=1}^n e_1\wedge \cdots \wedge B_{V, e_k} \wedge \cdots \wedge e_n \right) , \end{aligned}$$

and we can write

$$\begin{aligned} B_{V, e_k} = \sum _{j=1}^n \langle B_{V, e_k} , Je_j\rangle Je_j. \end{aligned}$$

Now pick the frame field at $x_0$ to be

$$\begin{aligned} e_k = {1\over \sqrt{1+\lambda _k^2}} (\epsilon _k +\lambda _k J\epsilon _k), \qquad {\mathrm{for}} \ \ k=1,\ldots , n, \end{aligned}$$

so that at $x_0$ the vectors $e_k$ and $Je_k$ lie in the $\hbox {k}$th complex coordinate line. Recall that at the point p, $\Phi = dz_1\wedge \cdots \wedge dz_n$. Hence, at p

$$\begin{aligned} \begin{aligned} V e^{i\theta }&= e^{i\theta } i V\theta = \{ dz\}\left( \sum _{k=1}^n e_1\wedge \cdots \wedge B_{V, e_k} \wedge \cdots \wedge e_n \right) \\&= \sum _{k} dz_1(e_1) \cdots dz_k\left( \sum _j \langle B_{V, e_k} , Je_j\rangle Je_j \right) \cdots dz_n(e_n) \\&= \sum _{k} dz_1(e_1) \cdots dz_k\left( \langle B_{V, e_k} , Je_k\rangle Je_k \right) \cdots dz_n(e_n) \\&\equiv \sum _{k} dz_1(e_1) \cdots dz_k\left( \alpha _k Je_k \right) \cdots dz_n(e_n) \qquad {\mathrm{with}}\ \ \alpha _k \equiv \langle B_{V, e_k} , Je_j\rangle \\&= \sum _{k} dz_1(e_1) \cdots i \alpha _k dz_k\left( e_k \right) \cdots dz_n(e_n) \\&= i \sum _k \alpha _k dz(e_1\wedge \cdots \wedge e_n) = i \left( \sum _k \alpha _k\right) e^{i\theta }. \end{aligned} \end{aligned}$$

Hence, with summation convention,

$$\begin{aligned} \begin{aligned} V \theta&=\sum _k \alpha _k = \langle B_{V, e_k} , Je_k\rangle = \langle \nabla _{e_k} V, Je_k\rangle \\&= -\langle V, \nabla _{e_k} Je_k\rangle = - \langle V, J\nabla _{e_k} e_k\rangle = \langle JV, B_{e_k, e_k}\rangle \\&= \langle JV, H\rangle . \end{aligned} \end{aligned}$$

$\square$

Appendix B. The sharpness of our boundary convexity

Part of the point of this appendix is to show that strict $\overrightarrow{\mathbf{F}}_\theta$-convexity of the boundary $\partial \Omega$ is the right, i.e., borderline condition for the Dirichlet problem. This goes back to work in [2] for $\theta$ in the highest interval. We also discuss how this convexity relates to convexity of the domain $\Omega$.

Let

• $\Omega \subset \subset {{\mathbb {R}}}^n$ be a domain with smooth boundary $\partial \Omega$,

• $\mathbf{F}\subset {\mathrm{Sym}}^2({{\mathbb {R}}}^n)$ be a subequation,

• $\lambda >0$ be such that $(-{{{\mathcal {P}}}}) \cap (\lambda I+ \mathbf{F}) = \emptyset$ (this always exists), and

• $\varphi = -{\lambda \over 2} \Vert x\Vert ^2\bigr |_{\partial \Omega }$.

Definition B.1

$\Omega$ is said to be $\mathbf{F}(\Omega )$-convex if for every $K\subset \subset \Omega$, one has ${\widehat{K}}^{\mathbf{F}(\Omega )}\subset \subset \Omega$ where ${\widehat{K}}^{\mathbf{F}(\Omega )}{\mathop {=}\limits ^{{\mathrm{def}}}}\{x\in \Omega : f(x) \le \sup _K f, \ \forall \, f\in \mathbf{F}(\Omega )\}$

Theorem B.2

Assume there exists a function $h\in C^2({\overline{\Omega }})$ such that

• (i) $h\bigr |_{\Omega } \in \mathbf{F}(\Omega )$,

• (ii) $h\bigr |_{\partial \Omega } = \varphi$.

Then the domain $\Omega$ is $\mathbf{F}$-convex.

Furthermore, let $h^\star {\mathop {=}\limits ^{{\mathrm{def}}}}h + {\lambda \over 2} \Vert x\Vert ^2$. Then for every $x\in \partial \Omega$ where $(Dh^\star )_x \ne 0$, there exits $A\in ( \lambda I + \mathbf{F})$ such that

$$\begin{aligned} A\bigr |_{T_x(\partial \Omega )} = c_x B_x \end{aligned}$$

(B.1)

where $c_x>0$ and $B_x$ is the second fundamental form of $\partial \Omega$ at x with respect to the interior normal.

In particular, if $\mathbf{F}= \overrightarrow{{\mathbb {G}}}$ for a subequation ${{\mathbb {G}}}$, then $\partial \Omega$ is strictly ${{\mathbb {G}}}$-convex.

Proof

We begin by proving the second assertion. Fix $p\in \partial \Omega$ and w.l.o.g. assume p is the origin. Choose coordinates $x= (x', x_n)$ such that in a neighborhood of 0

$$\begin{aligned} \Omega = \{(x', x_n) : x_n \ge g(x')\} \end{aligned}$$

where g is $C^\infty$ with

$$\begin{aligned} g(0) = 0, \qquad (Dg)_0 = 0, \qquad {\mathrm{and}}\qquad (D^2 g)_0 = B \end{aligned}$$

where $B \ {\mathop {=}\limits ^{{\mathrm{def}}}}\ \text {the second fundamental form of }\partial \Omega \text { at }0 \text {w.r.t. the interior normal}$.

In a neighborhood U of we have a defining function for $\Omega$ given by

$$\begin{aligned} \rho (x) \equiv g(x')- x_n \end{aligned}$$

(1)

with

$$\begin{aligned} (D^2 \rho )_0 \ \equiv \ B. \end{aligned}$$

Lemma B.3

If $\widetilde{\rho }$ is any other defining function for $\Omega$ in U, then

$$\begin{aligned} (D^2\widetilde{\rho })\bigr |_{T_0(\partial \Omega )} \ \equiv \ |D\widetilde{\rho }|_0 B \end{aligned}$$

Proof

In a neighborhood of 0 we have that $\widetilde{\rho }(x) = a(x)\rho (x)$ where $a>0$. Now

$$\begin{aligned}&D\widetilde{\rho }= (Da)\rho + a (D\rho )\\&D^2\widetilde{\rho }= (D^2 a)\rho + (Da)\circ (D\rho ) + a(D^2 \rho ). \end{aligned}$$

At $x=0$ we have $\rho (0)=0$ and $(D\rho )_0=(0,\ldots , 0, -1) \equiv n$. Therefore,

$$\begin{aligned}&(D\widetilde{\rho })_0 = (0,\ldots , 0 , -a(0))\\&(D^2\widetilde{\rho })_0 = (Da \circ n) + a(0)(D^2 \rho )_0 \qquad \text {and so}\\&(D^2\widetilde{\rho })\bigr |_{T_0(\partial \Omega )} = a(0)(D^2 \rho )\bigr |_{T_0(\partial \Omega )} = a(0) B. \end{aligned}$$

$\square$

Now $\varphi = -{\lambda \over 2}\Vert x\Vert |^2\bigr |_{\partial \Omega }$, and $h\in C^2({\overline{\Omega }})$ is $\mathbf{F}$-subharmonic on $\Omega$ with boundary values $\varphi$. That is,

• (1) $D^2 h \ \in \ \mathbf{F}$ on ${\overline{\Omega }}$,

• (2) $( h-\varphi )\bigr |_{\partial \Omega } = 0$.

Recall that $h^\star \ \equiv \ h + \frac{\lambda }{2} \Vert x\Vert ^2$ Note that $h^\star \in C^2({\overline{\Omega }})$ is an $\mathbf{F}^\star$-subharmonic function for the subequation

$$\begin{aligned} \mathbf{F}^\star \ {\mathop {=}\limits ^{{\mathrm{def}}}}\ \lambda I+\mathbf{F}\end{aligned}$$

with

$$\begin{aligned} h^\star \bigr |_{\partial \Omega } = 0. \end{aligned}$$

Note also that $D^2 h^\star \ \in \ \lambda I+\mathbf{F}\ \subset \ {\mathrm{Int}}\mathbf{F}$ by the positivity condition. Hence, $h^\star$ is strictly $\mathbf{F}$-subharmonic.

With the supposition that $(Dh^\star )_x \ne 0$. we have that $h^\star$ is a defining function for $\partial \Omega$ in a neighborhood of x. We now apply Lemma B.3 to establish the second assertion.

Now for the first assertion. By the definition of $\lambda$ and Theorem 3.1 in [SMP] we know that $h^\star$ satisfies the Strong Maximum Principle, and so $h^\star <0$ on $\Omega$. Suppose $K\subset \subset \Omega$ and let $\delta \equiv {\mathrm{dist}}(K,\partial \Omega ) <0$. Then we have ${\widehat{K}}^{\mathbf{F}(\Omega )} \subset \Omega _\delta {\mathop {=}\limits ^{{\mathrm{def}}}}\{x\in \Omega : h^\star (x)\le \delta \}$. $\square$