Research

We introduce a (variation of quadrics) ansatz for constructing explicit, real-valued solutions to broad classes of complex Hessian equations on domains in \(\mathbb{C}^{n+1}\) and real Hessian equations on domains in \(\mathbb{R}^{n+1}\). In the complex setting, our method simultaneously addresses the deformed Hermitian-Yang-Mills/Leung-Yau-Zaslow (dHYM/LYZ) equation, the Monge-Ampère equation, and the J-equation. Under this ansatz each PDE reduces to a second-order system of ordinary differential equations admitting explicit first integrals. These ODE systems integrate in closed form via abelian integrals, producing wide families of explicit solutions together with a detailed description. In particular, on \(\mathbb{C}^3\), we construct entire dHYM/LYZ solutions of arbitrary subcritical phase, and on \(\mathbb{R}^3\) we produce entire special Lagrangian solutions of arbitrary subcritical phase. More generally, in any complex or real dimension, our ansatz yields entire solutions of certain subcritical phases for both the dHYM/LYZ and special Lagrangian equations. Some of these solutions develop singularities on compact regions. In the special Lagrangian case we show that, after a natural extension across the singular locus, these blow-up solutions coincide with previously known complete special Lagrangian submanifolds obtained via a different ansatz.

This paper explores the Bernstein problem of smooth maps \(f:\mathbb{R}^4\to\mathbb{R}^3\) whose graphs form coassociative submanifolds in \(\mathbb{R}^7\). We establish a condition, expressed in terms of its 2-dilations, that ensures \(f\) is affine. To the best of our current knowledge, our theorem achieves the minimal deficiency regarding the Lawson-Osserman cone. A corresponding result is also established for Cayley submanifolds in \(\mathbb{R}^8\).

It is known that minimal Lagrangians in Kähler-Einstein manifolds of non-positive scalar curvature are linearly stable under Hamiltonian deformations. We prove that they are also stable under the Lagrangian mean curvature flow, and therefore establish the equivalence between linear stability and dynamical stability.
Specifically, if one starts the mean curvature flow with a Lagrangian which is \(\mathcal{C}^1\)-close and Hamiltonian isotopic to a minimal Lagrangian, the flow exists smoothly for all time, and converges to that minimal Lagrangian. Due to the work of Neves [Ann. of Math. (2013)], this cannot be true for \(\mathcal{C}^0\)-closeness.

In this paper, we construct solutions of Lagrangian mean curvature flow which exist and are embedded for all time, but form an infinite-time singularity and converge to an immersed special Lagrangian as \(t\to\infty\). In particular, the flow decomposes the initial data into a union of special Lagrangians intersecting at one point. This result shows that infinite-time singularities can form in the Thomas-Yau [Comm. Anal. Geom. (2002),] `semi-stable' situation. A precise polynomial blow-up rate of the second fundamental form is also shown.
The infinite-time singularity formation is obtained by a perturbation of an approximate family \(N^{\varepsilon(t)}\) constructed by gluing in special Lagrangian `Lawlor necks' of size \(\varepsilon(t)\), where the dynamics of the neck size \(\varepsilon(t)\) are driven by the obstruction for the existence of nearby special Lagrangians to \(N^{\varepsilon(t)}\). This is inspired by the work of Brendle and Kapouleas [Comm. Pure Appl. Math. (2017),] regarding ancient solutions of the Ricci flow.

Given an entire \(\mathcal{C}^2\) function \(u\) on \(\mathbb{R}^n\), we consider the graph of \(Du\) as a Lagrangian submanifold of \(\mathbb{R}^{2n}\), and deform it by the mean curvature flow in \(\mathbb{R}^{2n}\). This leads to the special Lagrangian evolution equation, a fully nonlinear Hessian type PDE. We prove longtime existence and convergence results under a 2-positivity assumption of \(({\bf I}+(D^2 u)^2)^{-1}D^2 u\). Such results were previously known only under the stronger assumption of positivity of \(D^2u\).

We prove regularity, global existence, and convergence of Lagrangian mean curvature flows in the two-convex case. Such results were previously only known in the convex case, of which the current work represents a significant improvement. The proof relies on a newly discovered monotone quantity that controls two-convexity. Through a unitary transformation, same result for the mean curvature flow of area-decreasing Lagrangian submanifolds were established.

A new monotone quantity in graphical mean curvature flows of higher codimensions is identified in this work. The submanifold deformed by the mean curvature flow is the graph of a map between Riemannian manifolds, and the quantity is monotone increasing under the area-decreasing condition of the map. The flow provides a natural homotopy of the corresponding map and leads to sharp criteria regarding the homotopic class of maps between complex projective spaces, and maps from spheres to complex projective spaces, among others.

In this note, we study submanifold geometry of the Atiyah-Hitchin manifold, a double cover of the 2-monopole moduli space, which plays an important role in various settings such as the supersymmetric background of string theory. When the manifold is naturally identified as the total space of a line bundle over \(\mathbb{S}^2\), the zero section is a distinguished minimal 2-sphere of considerable interest. In particular, there has been a conjecture [Micallef and Wolfson, Math. Ann. (1993), Remark on p.262] about the uniqueness of this minimal 2-sphere among all closed minimal 2-surfaces. We show that this minimal 2-sphere satisfies the `strong stability condition' proposed in our earlier work [Tsai and Wang, J. Reine Angew. Math. (2020)], and confirm the global uniqueness as a corollary.

We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a \(\mathcal{C}^1\) dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a \(\mathcal{C}^1\) neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem [Tsai and Wang, J. Differential Geom. (2018)] which applies only to calibrated submanifolds of special holonomy ambient manifolds.

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant-Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean curvature flow. The proof relies on intricate interconnections of the Ricci flatness of the ambient space and the extrinsic geometry of the calibrated submanifolds.

We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered cohomologies give a two-sided resolution of Lefschetz maps, and thereby, they are directly related with the kernels and cokernels of the Lefschetz maps. We also introduce a novel, non-associative product operation on differential forms for symplectic manifolds. This product generates an \(A_{\infty}\)-algebra structure on forms that underlies the filtered cohomologies and gives them a ring structure. As an application, we demonstrate how the ring structure of the filtered cohomologies can distinguish different symplectic four-manifolds in the context of a circle times a fibered three-manifold.

Given an open book decomposition \( (\Sigma,\tau) \)of a three manifold \(Y\), Thurston and Winkelnkemper [Proc. Amer. Math. Soc. (1975)] construct a specific contact form \(a\) on \(Y\). Given a spin-c Dirac operator \(\mathcal{D}\) on \(Y\), the contact form naturally associates a one parameter family of Dirac operators \(\mathcal{D}_r = \mathcal{D} - \frac{ir}{2}{\rm cl}(a)\) for \(r\geq0\). When \(r>\!>1\), we prove that the spectrum of \(\mathcal{D}_r = \mathcal{D}_0 - \frac{ir}{2}{\rm cl}(a)\) within \( [-\frac{1}{2}r^{\frac{1}{2}},\frac{1}{2}r^{\frac{1}{2}}]\) are almost uniformly distributed. With the result in Part I [J. Symplectic Geom. (2017)], it implies that the subleading order term of the spectral flow from \(\mathcal{D}_0\) to \(\mathcal{D}_r\) is of order \(r(\log r)^{\frac{9}{2}}\). Besides the interests of the spectral flow, the method of this paper provide a tool to analyze the Dirac operator on an open book decomposition.

Let \(Y\) be a compact, oriented 3-manifold with a contact form \(a\) and a metric \({\rm d}s^2\). Suppose that \(F\to Y\) is a principal bundle with structure group \({\rm U}(2) = {\rm SU}(2)\times_{\pm1}\mathbb{S}^1\) such that \(F/\mathbb{S}^1\) is the principal \({\rm SO}(3)\) bundle of orthonormal frames for \(TY\). A unitary connection \(A_0\) on the Hermitian line bundle \(F\times_{\det{\rm U}(2)}\mathbb{C}\) determines a self-adjoint Dirac operator \(\mathcal{D}_0\) on the \(\mathbb{C}^2\)-bundle \(F\times_{{\rm U}(2)}\mathbb{C}^2\).
The contact form \(a\) can be used to perturb the connection \(A_0\) by \(A_0 - ira\). This associates a one parameter family of Dirac operators \(\mathcal{D}_r\) for \(r\geq0\). When \(r>\!>1\), we establish a sharp sup-norm estimate on the eigensections of \(\mathcal{D}_r\) with small eigenvalues. The sup-norm estimate can be applied to study the asymptotic behavior of the spectral flow from \(\mathcal{D}_0\) to \(\mathcal{D}_r\). In particular, it implies that the subleading order term of the spectral flow is strictly smaller than \(O(r^{\frac{3}{2}})\). We also relate the \(\eta\)-invariant of \(\mathcal{D}_r\) to certain spectral asymmetry function involving only the small eigenvalues of \(\mathcal{D}_r\).

The phase space of a particle or a mechanical system contains an intrinsic symplectic structure, and hence, it is a symplectic manifold. Recently, new invariants for symplectic manifolds in terms of cohomologies of differential forms have been introduced by Tseng and Yau. Here, we discuss the physical motivation behind the new symplectic invariants and analyze these invariants for phase space, i.e. the non-compact cotangent bundle.

Let \(Y\) be a compact, oriented 3-manifold with a contact form \(a\). For any Dirac operator \(\mathcal{D}\), we study the asymptotic behavior of the spectral flow between \(\mathcal{D}\) and \(\mathcal{D} - \frac{ir}{2}{\rm cl}(a)\) as \(r\to\infty\). If \(a\) is the Thurston-Winkelnkemper contact form whose monodromy is the product of Dehn twists along disjoint circles, we prove that the next order term of the spectral flow function is \(O(r)\).