Mean curvature flow of monotone Lagrangian submanifolds

We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in $\mathbb C^{n}$.Comment: 37 pages, 3 figure

Singularities of Lagrangian mean curvature flow: zero-Maslov class case

We study singularities of Lagrangian mean curvature flow in \C^n when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonian isotopic to a plane that, nevertheless, develop finite time singularities under mean curvature flow. We then prove two theorems regarding the tangent flow at a singularity when the initial condition is a zero-Maslov class Lagrangian. The first one (Theorem A) states that that the rescaled flow at a singularity converges weakly to a finite union of area-minimizing Lagrangian cones. The second theorem (Theorem B) states that, under the additional assumptions that the initial condition is an almost-calibrated and rational Lagrangian, connected components of the rescaled flow converges to a single area-minimizing Lagrangian cone, as opposed to a possible non-area-minimizing union of area-minimizing Lagrangian cones. The latter condition is dense for Lagrangians with finitely generated $H_1(L,\Z)$.Comment: 34 pages. 3 figures. To appear in Inventione

Mean Curvature Flow of Spacelike Graphs

We prove the mean curvature flow of a spacelike graph in $(\Sigma_1\times \Sigma_2, g_1-g_2)$ of a map $f:\Sigma_1\to \Sigma_2$ from a closed Riemannian manifold $(\Sigma_1,g_1)$ with $Ricci_1> 0$ to a complete Riemannian manifold $(\Sigma_2,g_2)$ with bounded curvature tensor and derivatives, and with sectional curvatures satisfying $K_2\leq K_1$, remains a spacelike graph, exists for all time, and converges to a slice at infinity. We also show, with no need of the assumption $K_2\leq K_1$, that if $K_1>0$, or if $Ricci_1>0$ and $K_2\leq -c$, $c>0$ constant, any map $f:\Sigma_1\to \Sigma_2$ is trivially homotopic provided $f^*g_2<\rho g_1$ where $\rho=\min_{\Sigma_1}K_1/\sup_{\Sigma_2}K_2^+\geq 0$, in case $K_1>0$, and $\rho=+\infty$ in case $K_2\leq 0$. This largely extends some known results for $K_i$ constant and $\Sigma_2$ compact, obtained using the Riemannian structure of $\Sigma_1\times \Sigma_2$, and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo-Riemannian setting then in the Riemannian one.Comment: version 5: Math.Z (online first 30 July 2010). version 4: 30 pages: we replace the condition $K_1\geq 0$ by the the weaker one $Ricci_1\geq 0$. The proofs are essentially the same. We change the title to a shorter one. We add an applicatio

Energy properness and Sasakian-Einstein metrics

In this paper, we show that the existence of Sasakian-Einstein metrics is closely related to the properness of corresponding energy functionals. Under the condition that admitting no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of Sasakian-Einstein metric implies a Moser-Trudinger type inequality. At the end of this paper, we also obtain a Miyaoka-Yau type inequality in Sasakian geometry.Comment: 27 page

Impacts of soil conditions and light availability on natural regeneration of Norway spruce Picea abies (L.) H. Karst. in low-elevation mountain forests

& Key message Natural regeneration of P. abies (L.) H. Karst. may reach high densities in lower mountain elevations. The highest densities were found in sites with moderate light availability, with low pH, and not near the riverbank. However, age-height classes differed in the predicted magnitude of response, but were consistent in response directions. Mosses and understory species typical of coniferous forests were positively correlated with regeneration density. & Context Norway spruce Picea abies (L.) H. Karst. in Central Europe is at risk under climate change scenarios, particularly in mountain regions. Little is known about the impact of environmental factors on the natural regeneration of P. abies in lowelevation mountain forests. & Aims We aimed to assess impacts of distance from the riverbank, soil pH, and light availability on natural P. abies regeneration. We hypothesized that (1) natural P. abiesregeneration would depend on light availability and soil pH and (2) there are understory plant species which may indicate the microsites suitable for natural regeneration of P. abies. & Methods The study was conducted in the Stołowe Mountains National Park (SW Poland, 600–800 m a.s.l.). We established 160 study plots (25 m2 ) for natural regeneration, light availability, soil pH, and understory vegetation assessment

Ueber symmetrische Hyperflaechen in Riemannschen Mannigfaltigkeiten, die sich unter dem mittleren Kruemmungsfluss zu einer Lie-Gruppe zusammenziehen

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