In this paper we provide a criterion for the quasi-autonomous Hamiltonian path ("Hofer's geodesic") on arbitrary closed symplectic manifolds (M>,w) to be length minimizing in its homotopy class in terms of the spectral invariants p(G;1) that the author has recently constructed. As an application, we prove that any autonomous Hamiltonian path on arbitrary closed symplectic manifolds is length minimizing in its homotopy class with fixed ends, as long as it has no contractible periodic orbits of period one and it has a maximum and a minimum that are generically under-twisted, and all of its critical points are non-degenerate in the Floer theoretic sense.

Our point of departure is the following $L^{p}$ boundedness result form [St], Theorem 1.1, ... Recently, the results in Theorem 1.1 were improved by Fan, Guo, and Pan in [FGP] who showed that the $L^{p}$ boundedness of $T_{\Phi}$ and $M_{\Phi}$ continues to hold if the condition $\Omega \in \mathbb{C}^{1}(Sn-1)$ is replaced by the weaker condition $\Omega \in L^{q}(Sn-1)$ for some $q > 1$. Also, the authors of [FGP] were able to establish the $L^{p}$ boundedness of the maximal operator $T^{*}_{\Phi}$ under the condition $\Omega \in L^{q}(Sn-1)$ for some $q > 1$. The main purpose of this paper is to present further improvements of the above results in which the condition $\Omega \in L^{q}(Sn-1)$ is replaced by a weaker condition $\Omega \in B^{0,0}_{q}(Sn-1)$ ...

We prove Berstein type theroms for minimal n-submanifolds in R^n+p with flat normal bundle under certain growth conditions for p greater than or equal to 2 and n less than or equal to 5, as well for arbitrary n and p = 1. When M is a graphic minimal hyperface we recover the known result.

We show how a ruled minimal Lagrangian submanifold of complex projective 3-space may be used to construct related minimal surfaces in the 5-sphere.

For any knot $T$ transverse to a given contact structure on a 3-manifold, we exhibit a Legendrian two-component link $\mathbb{L}=L^1\squarecup L^2$ such that $T$ equals the transverse push-off of $L_1$ and contact $(+1)$-surgery has the same effect as a Lutz twist along $T$.

We consider minimal closed hypersurfaces $M^4\subset \mathbb{S}^5(1)$ with constant scalar curvature. We prove that, if $M^4$ is additionally a Willmore hypersurface, then it is isoparametric. This gives a positive answer to the question made by Chern about the pinching of the scalar curvature for closed minimal Willmore hypersurfaces in dimension 4.

For all m greater than or equal to 1 we build a two-dimensional family of smooth manifolds of real dimension 3m + 2 and use it to interpolate between the anticanonical family in CP^{m +1} and its mirror dual. The main tool is the notion of self-dual manifolds.

We determine the structure of the cut locus of a class of tori of revolution, which includes the standard tori in 3-dimensional Euclidean space.

We give a reduction of the conjecture that for terminal projective threefolds whose anticanonical divisors are nef, the second Chern classes are pseudo-effective. On the other hand, some effective non-vanishing results are obtained as applications of the pseudo-effectivity of the second Chern classes.

We give the characterization of Arnold-Mather type for stable singular Legendre immersions in connection with the classification problem. The most important building block of the theory is providing a module structure on the space of infinitesimal integral deformations by means of the notion of natural liftings of differential systems and of contact Hamiltonian vector fields.