In this article we define new flows on the Hitchin components for PGL(V). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present in the case when n=2. We determine a global coordinate system on the Hitchin component. Using the computation of the Goldman symplectic form on the Hitchin component, that is developed by two of the authors in a companion paper to this article (Sun and Zhang in The Goldman symplectic form on the PGL(V)-Hitchin component, 2017. arXiv:1709.03589), this gives a global Darboux coordinate system on the Hitchin component.

In this article we show that for every finite area hyperbolic surface X of type (g,n) and any harmonic Beltrami differential μ on X , then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to -g as the genus g goes to infinity.

In this paper, we prove that for any Kähler metrics ω and χ on M, there exists a Kähler metric ω_φ = ω_0 + √-1 ∂ \bar∂ φ > 0 satisfying the J-equation tr_{ω_φ} χ = c if and only if (M, [ω0], [χ]) is uniformly J-stable. As a corollary, we find a sufficient condition for the existence of constant scalar curvature Kähler metrics with c_1 < 0. Using the same method, we also prove a similar result for the supercritical deformed Hermitian–Yang–Mills equation.

We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D∈|-KY| is a smooth divisor with D^2=d , then X=Y\D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y=P^2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type I_d fiber appearing as a singular fiber in a rational elliptic surface π: Y →P^1.

If 𝑈:[0,+∞[×𝑀 is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation ∂_𝑡𝑈+𝐻(𝑥,∂_𝑥 𝑈)=0, where 𝑀 is a not necessarily compact manifold, and 𝐻 is a Tonelli Hamiltonian, we prove the set Σ(𝑈), of points in ]0,+∞[×𝑀 where 𝑈 is not differentiable, is locally contractible. Moreover, we study the homotopy type of Σ(𝑈). We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.