Degenerate Special Lagrangian Equation (DSL)

• DSL is a degenerate elliptic, fully nonlinear phase equation defined on a space-time domain, modeling geodesics in the space of positive graph Lagrangians.
• It utilizes a lifted space-time Lagrangian angle and Harvey–Lawson subequation branches to rigorously handle its intrinsic degeneracy and singular set.
• The DSL framework underpins weak geodesic theory and connects with obstacle formulations and dHYM geodesic equations, influencing calibrated geometry studies.

The degenerate special Lagrangian equation (DSL) is a degenerate elliptic, fully nonlinear phase equation on a space-time domain $\mathcal D=(0,1)\times D\subset \mathbb R^{n+1}$, introduced as the PDE governing geodesics in the space of positive graph Lagrangians in $\mathbb C^n$. In its classical Euclidean form, for phase $\theta\in(-\pi,\pi]$ and $u\in C^2(\mathcal D)$, it is written

$$\operatorname{Im}\left(e^{-\sqrt{-1}\theta}\det(I_n+\sqrt{-1}\nabla^2 u)\right)=0, \qquad \operatorname{Re}\left(e^{-\sqrt{-1}\theta}\det(I+\sqrt{-1}\nabla_x^2 u)\right)>0,$$

where $I_n=\operatorname{diag}(0,1,\dots,1)$. The zero in the time-time slot makes the operator intrinsically degenerate, and much of the theory is organized around its lifted space-time Lagrangian angle, its Harvey–Lawson subequation branches, and its role in weak geodesic theory for positive Lagrangians (Rubinstein et al., 2015, Dellatorre, 2017).

1. Definition, lifted angle, and branch structure

Let $D\subset \mathbb R^n$ be bounded and set $\mathcal D=(0,1)\times D$. The matrix

$I_n=\operatorname{diag}(0,1,\dots,1)$

distinguishes the DSL from the ordinary $(n+1)$-dimensional special Lagrangian equation: the time direction is not weighted by a positive identity entry. This is why the equation is degenerate rather than uniformly elliptic (Rubinstein et al., 2015).

For $\mathbb C^n$0, the space-time Lagrangian angle is initially defined where the determinant is nonzero by

$\mathbb C^n$1

To formulate a global branch condition, Rubinstein–Solomon introduced a lifted angle on symmetric matrices. Writing

$\mathbb C^n$2

the lifted space-time angle can be expressed as

$\mathbb C^n$3

where $\mathbb C^n$4 is the ordinary lifted Lagrangian angle in $\mathbb C^n$5 variables (Pidaparthy et al., 22 Jul 2025).

The associated DSL subequation is

$\mathbb C^n$6

This branch formulation is essential because $\mathbb C^n$7 does not extend continuously across the singular set; only an upper semicontinuous extension exists, and the DSL branches are genuinely superlevel sets of that upper semicontinuous function (Dellatorre, 2017). In the Harvey–Lawson framework, $\mathbb C^n$8 is a subequation and its dual satisfies

$\mathbb C^n$9

A $\theta\in(-\pi,\pi]$0 solution of DSL is equivalently a function whose Hessian lies on the boundary of the chosen branch, together with the positivity condition selecting the admissible spatial phase interval (Rubinstein et al., 2015).

A common misconception is to identify DSL with the ordinary special Lagrangian equation in one higher dimension. That identification is incorrect: the operator uses $\theta\in(-\pi,\pi]$1, not the full identity, and the singular set $\theta\in(-\pi,\pi]$2 is a structural feature rather than a removable artifact.

2. Geometric origin in positive Lagrangians and geodesics

The geometric origin of DSL is the space of positive Lagrangians in an almost Calabi–Yau or Calabi–Yau manifold. If $\theta\in(-\pi,\pi]$3 is almost Calabi–Yau and $\theta\in(-\pi,\pi]$4 is Lagrangian, then $\theta\in(-\pi,\pi]$5 is $\theta\in(-\pi,\pi]$6-positive when

$\theta\in(-\pi,\pi]$7

On a Hamiltonian isotopy class of $\theta\in(-\pi,\pi]$8-positive Lagrangians, Solomon defined a weak Riemannian metric, and the geodesic equation is $\theta\in(-\pi,\pi]$9 for the velocity functions $u\in C^2(\mathcal D)$0 (Dellatorre, 2017).

In the Euclidean graph setting, one takes

$u\in C^2(\mathcal D)$1

For a graph Lagrangian $u\in C^2(\mathcal D)$2, the ordinary Lagrangian angle is

$u\in C^2(\mathcal D)$3

and positivity is the condition

$u\in C^2(\mathcal D)$4

Rubinstein–Solomon showed that a path $u\in C^2(\mathcal D)$5 is a geodesic in the space of positive graph Lagrangians if and only if $u\in C^2(\mathcal D)$6 satisfies the DSL with the slice positivity condition and the prescribed endpoint data (Rubinstein et al., 2015).

This identification persists beyond Euclidean space. Dellatorre proved that the Euclidean DSL induces a global Riemannian subequation on $u\in C^2(\mathcal D)$7 for every Riemannian manifold $u\in C^2(\mathcal D)$8, locally modeled on the Euclidean branch. The globalization uses Harvey–Lawson’s Riemannian $u\in C^2(\mathcal D)$9-subequation formalism with $$\operatorname{Im}\left(e^{-\sqrt{-1}\theta}\det(I_n+\sqrt{-1}\nabla^2 u)\right)=0, \qquad \operatorname{Re}\left(e^{-\sqrt{-1}\theta}\det(I+\sqrt{-1}\nabla_x^2 u)\right)>0,$$0, together with the $$\operatorname{Im}\left(e^{-\sqrt{-1}\theta}\det(I_n+\sqrt{-1}\nabla^2 u)\right)=0, \qquad \operatorname{Re}\left(e^{-\sqrt{-1}\theta}\det(I+\sqrt{-1}\nabla_x^2 u)\right)>0,$$1-invariance of the DSL branch (Dellatorre, 2017).

The geometric consequences are explicit in two settings. If $$\operatorname{Im}\left(e^{-\sqrt{-1}\theta}\det(I_n+\sqrt{-1}\nabla^2 u)\right)=0, \qquad \operatorname{Re}\left(e^{-\sqrt{-1}\theta}\det(I+\sqrt{-1}\nabla_x^2 u)\right)>0,$$2 is integrably parallelizable, then $$\operatorname{Im}\left(e^{-\sqrt{-1}\theta}\det(I_n+\sqrt{-1}\nabla^2 u)\right)=0, \qquad \operatorname{Re}\left(e^{-\sqrt{-1}\theta}\det(I+\sqrt{-1}\nabla_x^2 u)\right)>0,$$3 admits a Calabi–Yau structure and the Riemannian DSL on $$\operatorname{Im}\left(e^{-\sqrt{-1}\theta}\det(I_n+\sqrt{-1}\nabla^2 u)\right)=0, \qquad \operatorname{Re}\left(e^{-\sqrt{-1}\theta}\det(I+\sqrt{-1}\nabla_x^2 u)\right)>0,$$4 governs geodesics of positive graph Lagrangians in $$\operatorname{Im}\left(e^{-\sqrt{-1}\theta}\det(I_n+\sqrt{-1}\nabla^2 u)\right)=0, \qquad \operatorname{Re}\left(e^{-\sqrt{-1}\theta}\det(I+\sqrt{-1}\nabla_x^2 u)\right)>0,$$5. If $$\operatorname{Im}\left(e^{-\sqrt{-1}\theta}\det(I_n+\sqrt{-1}\nabla^2 u)\right)=0, \qquad \operatorname{Re}\left(e^{-\sqrt{-1}\theta}\det(I+\sqrt{-1}\nabla_x^2 u)\right)>0,$$6 is a smooth Calabi–Yau torus fibration over $$\operatorname{Im}\left(e^{-\sqrt{-1}\theta}\det(I_n+\sqrt{-1}\nabla^2 u)\right)=0, \qquad \operatorname{Re}\left(e^{-\sqrt{-1}\theta}\det(I+\sqrt{-1}\nabla_x^2 u)\right)>0,$$7, then the Riemannian DSL on $$\operatorname{Im}\left(e^{-\sqrt{-1}\theta}\det(I_n+\sqrt{-1}\nabla^2 u)\right)=0, \qquad \operatorname{Re}\left(e^{-\sqrt{-1}\theta}\det(I+\sqrt{-1}\nabla_x^2 u)\right)>0,$$8 governs geodesics in the space of positive Lagrangian sections in $$\operatorname{Im}\left(e^{-\sqrt{-1}\theta}\det(I_n+\sqrt{-1}\nabla^2 u)\right)=0, \qquad \operatorname{Re}\left(e^{-\sqrt{-1}\theta}\det(I+\sqrt{-1}\nabla_x^2 u)\right)>0,$$9 (Dellatorre, 2017). This shows that DSL is not confined to flat graph geometry; it is a global equation on Riemannian bases with direct geometric meaning.

3. Subequations, Dirichlet theory, and weak solutions

The analytic framework for DSL is Harvey–Lawson Dirichlet duality. For a subequation $I_n=\operatorname{diag}(0,1,\dots,1)$0, one studies $I_n=\operatorname{diag}(0,1,\dots,1)$1-subharmonicity in the viscosity sense, the dual subequation $I_n=\operatorname{diag}(0,1,\dots,1)$2, and $I_n=\operatorname{diag}(0,1,\dots,1)$3-harmonic functions satisfying

$I_n=\operatorname{diag}(0,1,\dots,1)$4

In the DSL setting, the branch $I_n=\operatorname{diag}(0,1,\dots,1)$5 is a pure second-order subequation, and the weak DSL equation is encoded as $I_n=\operatorname{diag}(0,1,\dots,1)$6-harmonicity together with the positivity/admissibility condition on spatial slices (Rubinstein et al., 2015).

A central technical issue is that the natural domain for the geodesic endpoint problem,

$I_n=\operatorname{diag}(0,1,\dots,1)$7

has corners. Rubinstein–Solomon extended Harvey–Lawson’s Perron method from smooth boundaries to manifolds with embedded corners and proved existence and uniqueness of continuous solutions to the $I_n=\operatorname{diag}(0,1,\dots,1)$8-Dirichlet problem in all branches, under the appropriate boundary convexity and admissibility hypotheses. In the concrete strictly convex Euclidean case, if the endpoint potentials satisfy

$I_n=\operatorname{diag}(0,1,\dots,1)$9

then there is a unique continuous solution with those endpoint values and side boundary data affine in $D\subset \mathbb R^n$0 (Rubinstein et al., 2015).

Dellatorre extended this corners theory to the Riemannian setting. If $D\subset \mathbb R^n$1 is a subequation on a Riemannian manifold $D\subset \mathbb R^n$2, comparison holds, and $D\subset \mathbb R^n$3 is strictly $D\subset \mathbb R^n$4-convex and strictly $D\subset \mathbb R^n$5-convex, then the $D\subset \mathbb R^n$6-Dirichlet problem on a bounded domain with embedded corners admits a unique solution in $D\subset \mathbb R^n$7. Applied to the Riemannian DSL on $D\subset \mathbb R^n$8, this yields continuous solutions under comparison, boundary convexity for the spatial special Lagrangian subequations, and admissible endpoint data (Dellatorre, 2017).

In important cases these assumptions are automatic. Dellatorre records that if $D\subset \mathbb R^n$9 is simply-connected and complete and $\mathcal D=(0,1)\times D$0 is bounded and strictly convex, then comparison holds for the Riemannian DSL and the required boundary convexity hypotheses follow from strict convexity of $\mathcal D=(0,1)\times D$1. The resulting solution is unique in $\mathcal D=(0,1)\times D$2 (Dellatorre, 2017).

Rubinstein–Solomon also obtained additional structure in the outer branches. Time-slice estimates imply Lipschitz control in $\mathcal D=(0,1)\times D$3, interior slices satisfy the ordinary special Lagrangian subequation $\mathcal D=(0,1)\times D$4, and in the outer branches the solutions are $\mathcal D=(0,1)\times D$5 overall. They further introduced a calibration measure

$\mathcal D=(0,1)\times D$6

which extends continuously to convex or concave functions and is positive in the outer branches. This measure allows one to define the length integral of weak geodesics in those branches (Rubinstein et al., 2015).

4. Minimum principle, partial Legendre transform, and obstacle formulations

Darvas–Rubinstein established a minimum principle for the top branch of DSL. If

$\mathcal D=(0,1)\times D$7

then

$\mathcal D=(0,1)\times D$8

This is the Lagrangian analogue of the classical minimum principle in convex analysis and the Kiselman principle in complex analysis (Darvas et al., 2016).

The mechanism is an exact relation between the space-time angle and the spatial angle after minimization. In the smooth, strictly convex-in-$\mathcal D=(0,1)\times D$9 case, if $I_n=\operatorname{diag}(0,1,\dots,1)$0 and $I_n=\operatorname{diag}(0,1,\dots,1)$1 is the minimizer, then

$I_n=\operatorname{diag}(0,1,\dots,1)$2

At the Hessian level, the minimized function satisfies the Schur-complement formula

$I_n=\operatorname{diag}(0,1,\dots,1)$3

and consequently

$I_n=\operatorname{diag}(0,1,\dots,1)$4

These identities make precise the $I_n=\operatorname{diag}(0,1,\dots,1)$5 phase shift from space-time to space (Darvas et al., 2016).

The minimum principle leads to a representation formula for weak DSL solutions. For the Dirichlet problem on $I_n=\operatorname{diag}(0,1,\dots,1)$6, if $I_n=\operatorname{diag}(0,1,\dots,1)$7 is the unique weak solution and

$I_n=\operatorname{diag}(0,1,\dots,1)$8

is the time partial Legendre transform, then $I_n=\operatorname{diag}(0,1,\dots,1)$9 solves an obstacle problem for the ordinary special Lagrangian subequation $(n+1)$0. Equivalently, the space-time DSL problem can be reduced to a one-parameter family of spatial obstacle problems together with Legendre inversion (Darvas et al., 2016). This places DSL in close formal proximity to homogeneous real and complex Monge–Ampère geodesic theory.

5. Convexity, affine shears, and the top two branches

Recent work sharpened the branch-dependent convexity theory of DSL. For the top two branches

$(n+1)$1

the basic matrix estimate is that if

$(n+1)$2

then $(n+1)$3, and if $(n+1)$4, then $(n+1)$5. For $(n+1)$6 subsolutions this yields

$(n+1)$7

with strict positivity when the mixed derivative $(n+1)$8 does not vanish. This is a partial $(n+1)$9 estimate arising from branch admissibility alone (Pidaparthy et al., 22 Jul 2025).

The key new mechanism is a space-time affine shear

$\mathbb C^n$00

together with the transformed matrix $\mathbb C^n$01, whose spatial block is the pullback $\mathbb C^n$02. The decisive identity is

$\mathbb C^n$03

Thus affine shearing preserves the space-time Lagrangian angle, and every affine space-time slice of a DSL subsolution is an ordinary special Lagrangian subsolution one branch lower: $\mathbb C^n$04 This converts space-time information into spatial SL information on every nonvertical affine slice (Pidaparthy et al., 22 Jul 2025).

The resulting convexity theorem is branch-sharp. If $\mathbb C^n$05 with

$\mathbb C^n$06

then:

• $\mathbb C^n$07 is convex for every $\mathbb C^n$08;
• if $\mathbb C^n$09 is a 2-plane not containing a vertical line $\mathbb C^n$10, then $\mathbb C^n$11 is subharmonic;
• if

$\mathbb C^n$12

then $\mathbb C^n$13 is jointly convex in $\mathbb C^n$14.

This settles Rubinstein–Solomon’s question on joint space-time convexity in the top branch and shows that second-branch convexity is weaker but still structured (Pidaparthy et al., 22 Jul 2025).

The same paper proves the product decomposition

$\mathbb C^n$15

where $\mathbb C^n$16. This identifies the top two DSL branches as “one convex time direction plus a spatial special Lagrangian branch” in the sense of Ross–Witt Nyström. Using this structure, the minimum principle extends from the top branch to the second branch: $\mathbb C^n$17 The Legendre-duality description of the Dirichlet problem correspondingly extends from the top branch to the top two branches (Pidaparthy et al., 22 Jul 2025).

The limitations are equally explicit. The second branch does not imply spatial convexity and does not imply global 2-convexity in general; the paper provides a quadratic example with $\mathbb C^n$18, and for suitable parameters the example is not even 2-convex. This shows that the second-branch conclusions are sharp rather than provisional (Pidaparthy et al., 22 Jul 2025).

6. Relation to deformed Hermitian–Yang–Mills and adjacent equations

DSL belongs to a broader family of special-Lagrangian-type phase equations, but its degeneracy and geodesic role distinguish it sharply from nearby theories. Jacob–Yau studied the deformed Hermitian–Yang–Mills equation (dHYM) for a Hermitian metric on a holomorphic line bundle over a compact Kähler manifold. In their formulation, if $\mathbb C^n$19 and $\mathbb C^n$20 are the eigenvalues of the curvature endomorphism, the phase is

$\mathbb C^n$21

and dHYM is the nondegenerate elliptic equation $\mathbb C^n$22. They established a variational interpretation, uniqueness up to constant rescaling, a complete existence criterion on Kähler surfaces, and a line-bundle mean curvature flow. However, that theory does not study DSL, geodesics in spaces of positive Lagrangians, or a degenerate space-time equation arising from such geodesics (Jacob et al., 2014).

The relation becomes closer in Jacob’s work on weak geodesics for dHYM. There the geodesic equation in the space of dHYM potentials is reformulated as a degenerate elliptic equation on $\mathbb C^n$23,

$\mathbb C^n$24

with positivity condition on spatial slices. The time direction is degenerate because $\mathbb C^n$25 vanishes in the annulus direction, and the equation is treated via a Hermitian space-time angle operator, Harvey–Lawson subequations, and convexity of the highest branch. Under the supercritical phase assumption

$\mathbb C^n$26

Jacob obtained continuous weak geodesics with prescribed endpoints in the dHYM potential space (Jacob, 2019).

This comparison clarifies the position of DSL within special-Lagrangian-type PDE. DSL is the real, space-time, degenerate elliptic equation governing geodesics of positive Lagrangian graphs and, after globalization, geodesics of positive Lagrangians on suitable Riemannian manifolds (Rubinstein et al., 2015, Dellatorre, 2017). dHYM is a nondegenerate elliptic line-bundle phase equation, while the dHYM geodesic equation is a complex or Hermitian analogue of DSL rather than DSL itself (Jacob et al., 2014, Jacob, 2019).

A plausible implication is that DSL should be viewed less as an isolated degenerate PDE than as the canonical space-time member of a broader phase-operator hierarchy: ordinary special Lagrangian and dHYM describe static calibrated objects, while DSL and dHYM weak-geodesic equations describe the corresponding infinite-dimensional geodesic geometries.