We prove that Mark Gross’  topological Calabi-Yau compactifications can be made into symplectic compactifications. To
prove this we develop a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds
with singularities, which we call simple. This produces pairs of compact symplectic 6-manifolds homeomorphic to mirror pairs of Calabi-Yau 3-folds together with Lagrangian fibrations whose underlying integral affine structures are dual.
We develop some results from  on the positivity of direct image bundles in the particular case of a trivial ¯bration over a one-dimensional base. We also apply the results to study variations of KÄahler metrics.
Le N Q, Sesum N. The mean curvature at the first singular time of the mean curvature flow[J]. Annales De L Institut Henri Poincare-analyse Non Lineaire, 2010, 27(6): 1441-1459.
Gian Paolo Leonardi · Simon Masnou. Locality of the mean curvature of rectifiable varifolds. 2009.
Ulrich Menne. Some Applications of the Isoperimetric Inequality for Integral Varifolds. 2008.
Ulrich Menne. Second order rectifiability of integral varifolds of locally bounded first variation. 2008.
Le N Q. On the convergence of the Ohta-Kawasaki Equation to motion by nonlocal Mullins-Sekerka Law[J]. Siam Journal on Mathematical Analysis, 2009, 42(4): 1602-1638.
Menne U. A Sobolev Poincar\\\u0027e type inequality for integral varifolds[J]. Calculus of Variations and Partial Differential Equations, 2008: 369-408.
Simon Masnou · Giacomo Nardi. A coarea-type formula for the relaxation of a generalized elastica functional. 2011.
Scientific Computing Matthias Rogeraffiliated Withcentre For Analysis · T U Applications · Universitat Tubingen Reiner Schatzleaffiliated Withmathematische…. On a Modified Conjecture of De Giorgi. 2006.
Bretin E, Lachaud J, Oudet E, et al. Regularization of Discrete Contour by Willmore Energy[J]. Journal of Mathematical Imaging and Vision, 2011, 40(2): 214-229.
Menne U. Decay estimates for the quadratic tilt-excess of integral varifolds[J]. Archive for Rational Mechanics and Analysis, 2009, 204(1): 1-83.
The weak mean curvature is lower semicontinuous under weak
convergence of varifolds, that is, if ¹k ! ¹ weakly as varifolds
then k ~H¹ kLp(¹)· lim infk!1 k ~H¹k kLp(¹k). In contrast, if
Tk ! T weakly as integral currents, then ¹T may not have a
locally bounded ¯rst variation even if k ~H¹Tk
kL1(¹k) is bounded.
In 1999, Luigi Ambrosio asked the question whether lower semi-
continuity of the weak mean curvature is true when T is assumed
to be smooth. This was proved in [AmMa03] for p > n =
dim T in Rn+1 using results from [Sch04]. Here we prove this
in any dimension and codimension down to the desired exponent
p = 2. For p = n = 2, this corresponds to the Willmore functional.
In a forthcoming joint work [RoSch06], main steps of the pre-
sent article are used to prove a modi¯ed conjecture of De Giorgi
that the sum of the area and the Willmore functional is the ¡-limit
of a di®use Landau-Ginzburg approximation.
Given a hyperbolic surface with geodesic boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S
that start and end at @S perpendicularly are coordinates on the Teichm¨uller space T (S). We express the Weil-Petersson Poisson
structure of T (S) in this system of coordinates, and we prove that it limits pointwise to the piecewise-linear Poisson structure
defined by Kontsevich on the arc complex of S. At the same time, we obtain a formula for the first-order variation of the distance between two closed geodesics under Fenchel-Nielsen deformation.
We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, aspherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple identity component, then the local isometry orbits in M are roughly ¯bers of a¯ber bundle. A corollary is that if M has an open, dense, locally homogeneous subset, then M is locally homogeneous.