[Paper Review] The rigidity theorems for Lagrangian self shrinkers

Qi Ding, Y. L. Xin|arXiv (Cornell University)|Dec 12, 2011

Geometric Analysis and Curvature Flows6 references3 citations

TL;DR

This paper establishes rigidity theorems for Lagrangian self-shrinkers in pseudo-Euclidean space ℝ²ⁿⁿ and Euclidean space ℝ²ⁿ using an integral method based on the drift Laplacian operator ℒ. It proves that any entire smooth convex solution to the Lagrangian mean curvature flow self-shrinker equation must be a quadratic polynomial, removing prior decay assumptions and extending earlier results via a novel application of the drift Laplacian and weighted integration by parts.

ABSTRACT

By the integral method we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in $\R^{2n}_{n}$ with the indefinite metric $\sum_i dx_idy_i$ is flat. This result improves the previous ones in \cite{HW} and \cite{CCY} by removing the additional assumption in their results. In a similar manner, we reprove its Euclidean counterpart which is established in \cite{CCY}.

Motivation & Objective

• To establish rigidity theorems for Lagrangian self-shrinkers in pseudo-Euclidean space ℝ²ⁿⁿ with indefinite metric ∑dxᵢdyᵢ.
• To remove the decay condition on the Hessian of u required in prior works [9] and [1] for classifying entire solutions to the self-shrinker equation.
• To reprove the Euclidean case rigidity result using the drift Laplacian operator ℒ and integral methods, offering a new proof strategy.
• To show that the phase function Θ = ∑arctanλᵢ satisfies ℒΘ = 0, enabling the use of integral estimates to prove constancy.

Proposed method

• Introduces the drift Laplacian operator ℒ defined via the weighted measure e⁻¹⁴ˣ·Du, which governs the evolution of the logarithmic determinant of the Hessian φ = log det D²u.
• Applies integration by parts with a cutoff function η and weight e⁻¹⁴ˣ·Du to derive energy estimates for ∇φ and ∇Θ.
• Uses the identity ℒφ = 0 for φ = log det D²u in the pseudo-Euclidean case, derived from the self-shrinker equation (1.2), to apply maximum and integral methods.
• Establishes that ∫|∇φ|²η²ρ dμ ≤ C∫|∇η|²φ²ρ dμ, and under volume growth and decay conditions on |∇η|²/|x|², shows φ is constant.
• Reinterprets the phase function Θ = ∑arctanλᵢ in the Euclidean case as satisfying ℒΘ = 0, enabling the same integral method to prove Θ is constant.
• Employs barrier functions and decay conditions on the Hessian or subharmonic function Δv to show that solutions to logΔv = ½x·Dv − v must be quadratic polynomials.

Experimental results

Research questions

• RQ1 Does every entire smooth convex solution to the Lagrangian self-shrinker equation in ℝ²ⁿⁿ with indefinite metric must be a quadratic polynomial, without additional decay assumptions on the Hessian?
• RQ2 Can the rigidity result for entire Lagrangian graphs in ℝ²ⁿ be reproven using the drift Laplacian and integral methods rather than maximum principles?
• RQ3 What conditions on the growth of Δv or det D²u ensure that solutions to logΔv = ½x·Dv − v are quadratic polynomials?
• RQ4 Is the phase function Θ = ∑arctanλᵢ constant on entire Lagrangian self-shrinkers in ℝ²ⁿ, and can this be shown via the drift Laplacian?
• RQ5 Under what decay conditions on the gradient of a cutoff function does the integral method imply constancy of φ = log det D²u?

Key findings

• Any entire smooth convex solution u to the equation log det D²u = ½x·Du − u in ℝⁿ is a quadratic polynomial, without requiring decay of the Hessian, thus improving prior results in [9] and [1].
• The drift Laplacian operator ℒ satisfies ℒφ = 0 for φ = log det D²u, enabling the use of integral estimates to prove rigidity.
• The phase function Θ = ∑arctanλᵢ on Lagrangian graphs in ℝ²ⁿ satisfies ℒΘ = 0, and via integral estimates, it is shown to be constant, implying the solution is quadratic.
• For the equation logΔv = ½x·Dv − v, if Δv grows sufficiently fast at infinity (e.g., Δv ≥ C/(|x|²log|x|)), then v is a quadratic polynomial.
• The paper constructs a counterexample-like function φ(x) = log(2n−4) − 2log|x| for n ≥ 3 that satisfies the drift equation ℒφ = 0 on ℝⁿ∖{0}, showing the necessity of global regularity.
• The decay condition ∫ℝⁿ∖Br |Dη|²/|x|² e⁻ϕ e⁻|x|²/⁴eϕ → 0 as r→∞ is sufficient to conclude φ is constant, and this holds under mild growth assumptions on eϕ.