Yu Yuan
We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.
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In this paper, we prove some Bernstein type results for $n$-dimensional minimal Lagrangian graphs in quaternion Euclidean space $H^n\cong R^{4n}$. In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in $C^n$
Let Σbe a complete minimal Lagrangian submanifold of \C^n. We identify regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of Σlies in one of these regions, then Σis an affine space.
The aim of this expository article is to reveal the equivalence of Jörgens' Theorem on the two dimensional unimodular Hessian equation and Fu's Theorem on the two dimensional special Lagrangian equation.
We establish quadratic asymptotics for solutions to special Lagrangian equations with supercritical phases in exterior domains. The method is based on an exterior Liouville type result for general fully nonlinear elliptic equations toward constant asymptotics of bounded Hessian, and also certain rotation arguments toward Hessian bound. Our unified approach also leads to quadratic asymptotics for convex solutions to Monge-Ampère equations (previously known), quadratic Hessian equations, and inverse harmonic Hessian equations over exterior domains.
Bernstein theorems for nonlinear geometric PDEs
本文围绕极小曲面、Monge–Ampère、特殊 Lagrangian 方程及极小曲面系统，综述几何偏微分方程中的 Bernstein 问题，内容源自 2024 年欧洲微分几何博士学校讲座。
We show Bernstein type results for the entire self-shrinking solutions to Lagrangian mean curvature flow in $(\mathbb{R}^n\times\mathbb{R}^n, g_τ)$. The proofs rely on a priori estimates and barriers construction.
Using an rotation of Yuan, we observe that the gradient graph of any semiconvex function is a Liouville manifold, that is, does not admit bounded harmonic functions. As a corollary, we find that any entire solution of the fourth order Hamiltonian stationary equation with Lagrangian phase angle uniformly larger than the critical angle must be a quadratic.
We show that any global solution to the special Lagrangian equations with the phase larger than a critical value must be quadratic.
Special Lagrangian pair of pants
本文在 $T^n$ 的余切丛中构造了一般维数的特殊拉格朗日裤子。
We construct a family of Lagrangian submanifolds in the complex sphere with a SO(n)-invariance property. Among them we find those which are special Lagrangian with respect with the Calabi-Yau structure defined by the Stenzel metric.
We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for special Lagrangian equations with certain convexity are largely known by now.
We derive a Liouville type result for special Lagrangian equations with certain "convexity" and restricted linear growth assumptions on the solutions.
We derive a priori interior Hessian and gradient estimates for special Lagrangian equation of phase at least a critical value in dimension three.
We establish interior regularity for convex viscosity solutions of the special Lagrangian equation. Our result states that all such solutions are real analytic in the interior of the domain.
We prove the existence of non-smooth solutions to Special Lagrangian Equations in the non-convex case.
A constant rank theorem for special Lagrangian equations
本文证明了特殊拉格朗日方程和二次 Hessian 方程鞍点解的常秩定理，并由此得到次临界相位下特殊拉格朗日方程的 Liouville 型刚性结果。
A priori interior estimates for special Lagrangian curvature equations
本文针对特殊拉格朗日曲率方程，在临界相和凸情形下建立了先验内部曲率估计，并证明了任意常数相的先验内部梯度估计。
We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.
We extend Osserman's lemma on the generalized Gauss map of two-dimensional minimal graphs of higher codimension, construct a Jenkins-Serrin type special Lagrangian Scherk graph explicitly, and generalize Calabi's correspondence between minimal graphs and maximal graphs.
Special Lagrangians with multiple isolated singularities
该研究将Caffarelli-Hardt-Simon扰动论证推广到特殊拉格朗日量领域，证明正则特殊拉格朗日锥的桥接原理，进而得到具有多个给定孤立锥形奇点的特殊拉格朗日子流形的存在性。
This is a sequel to [1] and [2], which study the second boundary problem for special Lagrangian curvature potential equation. As consequences, we obtain the existence and uniqueness of the smooth uniformly convex solution by the method of continuity through carrying out a-priori estimate on the solutions.
Special Lagrangian submanifolds are submanifolds of a Calabi-Yau manifold calibrated by the real part of the holomorphic volume form. In this paper we use elliptic theory for edge-degenerate differential operators on singular manifolds to study the moduli space of deformations of special Lagrangian submanifolds with edge singularities. We obtain a general theorem describing the local structure of the moduli space. When the obstruction space vanishes the moduli space is a smooth, finite dimensional manifold.
McLean studied the deformations of compact special Lagrangian submanifolds, showing in particular that they come in moduli spaces whose dimension depends only on the topology of the submanifold. In this article we study the analogous problem for non-compact, "asymptotically conical" SL submanifolds, with respect to various "boundary conditions at infinity".
Semi-convex viscosity solutions of the special Lagrangian equation
本文研究特殊拉格朗日方程在近负相位、小半凸条件下的粘性解，证明其光滑性、内部导数与指数型黑塞估计，并给出尖锐性反例及次临界相位整解的新刘维尔定理。
Nonconvex Sublevel Sets for the Three-Dimensional Special Lagrangian Equation
针对三维特殊拉格朗日方程，在相位Θ∈(π/2,π)时构造特定凸区域，发现其零狄利克雷解存在非凸负次水平集。
Suppose $M_{1}$ and $M_{2}$ are two special Lagrangian submanifolds of $\Rtn$ with boundary that intersect transversally at one point $p$. The set $M_{1} \cup M_{2}$ is a singular special Lagrangian variety with an isolated singularity at the point of intersection. Suppose further that the tangent planes at the intersection satisfy an angle condition (which always holds in dimension $n=3$). Then, $M_{1} \cup M_{2}$ is regularizable; in other words, there exists a family of smooth, minimal Lagrangian submanifolds $M_α$ with boundary that converges to $M_{1} \cup M_{2}$ in a suitable topology. T…
The Neumann problem of special Lagrangian type equations
该研究针对带临界与超临界相位的特殊拉格朗日型方程Neumann问题，通过建立二阶一致先验估计、直接证明边界双法向导数估计等技术，结合连续性方法得到存在性结果，还解决了经典Neumann问题。
Hessian estimates for special Lagrangian equation by doubling
本文通过倍增法给出了特殊拉格朗日方程内部Hessian估计的新证明，不依赖Michael-Simon平均值不等式，扩展了Korevaar对极小超曲面梯度估计的点wise证明。
Doubling Argument of the Hessian Estimate for the Special Lagrangian Equation on General Phases with Constraints
本文建立加倍论证方法，在不依赖 Michael-Simon 均值不等式的情况下，得到带一般相位约束的特殊拉格朗日方程的 Hessian 估计，并建立了具独立意义的 Alexandrov 型定理。
In this paper, we prove some rigidity theorems for the entire 2-convex solutions of 2-Hessian equation in Euclidean space. As an application, we obtain a Bernstein type theorem for global special Lagrangian graphs.
Liouville type theorems and Hessian estimates for special Lagrangian equations
本文针对特殊Lagrange方程在特定“凸性”条件下证明了Liouville型定理，核心方法是通过Neumann-Poincaré不等式和均值不等式建立解的整体Hessian估计，并进一步推导了该条件及超临界相位下的内部Hessian估计。
Geometric regularity for semi-convex viscosity solutions to the special Lagrangian equation
本文证明三维次临界特殊拉格朗日方程半凸粘性解梯度图面积最小且光滑，并推广至更高维度，推进了Wang-Yuan猜想。
Solutions of the Special Lagrangian Equation near Infinity
本文研究特殊拉格朗日方程在无穷远处的渐近行为，借助修正的 Kelvin 变换以原点附近单个函数刻画余项，并给出其在奇偶维数下的正则性差异。
The Gibbons-Hawking ansatz provides a large family of circle-invariant hyperkaehler 4-manifolds, and thus Calabi-Yau 2-folds. In this setting, we prove versions of the Thomas conjecture on existence of special Lagrangian representatives of Hamiltonian isotopy classes of Lagrangians, and the Thomas-Yau conjecture on long-time existence of the Lagrangian mean curvature flow. We also make observations concerning closed geodesics, curve shortening flow and minimal surfaces.
For every odd natural number g=2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in C^3 over a closed Riemann surface of genus g, using a geometric PDE gluing method.
Geodesics of positive Lagrangians from special Lagrangians with boundary
研究正拉格朗日子流形空间中的测地线，通过柱形变换揭示其与特殊拉格朗日柱体的关系，并证明相关空间的开性，提供非对称维度下的测地线解例。
We show that the degenerate special Lagrangian equation, recently introduced by Rubinstein-Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth Calabi-Yau torus fibration are governed by the Riemannian DSL on the base manifold. We then develop their analytic techniques, specifically modifications of the Dirichlet duality theory of Harvey-Lawson, in the Ri…
In this paper, we establish the existence and uniqueness theorem of the exterior Dirichlet problem for special Lagrangian equations with prescribed asymptotic behavior at infinity.
Non $C^1$ solutions to the special Lagrangian equation
本研究构造了特殊拉格朗日方程的Lipschitz连续但不具备$C^1$光滑性的粘性解，丰富了该方程解的正则性相关研究结果。
In this paper, we study a class of special Lagrangian curvature potential equations and obtain the existence of smooth solutions for Dirichlet problem. The existence result is based on a priori estimates of global $C^{0}$, $C^{1}$ and $C^{2}$ norms of solutions under the assumption of existence of a subsolution.
We obtain a quantitative high order expansion at infinity of solutions for a family of fully nonlinear elliptic equations on exterior domain, refine the study of the asymptotic behavior of the Monge-Ampère equation, the special Lagrangian equation and other elliptic equations, and give the precise gap between exterior maximal (or minimal) gradient graph and the entire case.
Analytic continuation problems are notoriously ill-posed without additional regularizing constraints, even though every analytic function has a rigidity property of unique continuation from every curve inside the domain of analyticity. In fact, well known theorems, guarantee that every continuous function can be uniformly approximated by analytic functions (polynomials or rational functions, for example). We consider several analytic continuation problems with typical global boundedness constraints. All such problems exhibit a power law precision deterioration as one moves away from the source…
Failure of Interior Hölder and Gradient Estimates for the Subcritical Special Lagrangian Equation
针对n≥3的次临界特殊拉格朗日方程，研究人员构造显式整体实解析解，证明基于振荡的内部C^α估计失效，且该族解无共同内部连续性模。
The exterior Dirichlet problem for special Lagrangian equations
本文针对特殊拉格朗日方程的外狄利克雷问题建立存在唯一性定理，推广了前人工作并改进衰减率，还求解了内狄利克雷问题，将其应用于三维二次Hessian方程的对应问题。
Notes on generalized special Lagrangian equation
本文通过研究扭曲特殊拉格朗日方程，获得了具有正Lipschitz右端项的Hessian（商）方程的先验$C^{1,1}$估计，并由此证明了3维情形下$σ_2=f^2(x)$粘性解的内部$C^{2,α}$正则性。
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