A two-dimensional periodic Schr\"{o}dingier operator is associated with every Lagrangian torus in the complex projective plane ${\mathbb C}P^2$. Using this operator we introduce an energy functional on the set of Lagrangian tori. It turns out this energy functional coincides with the Willmore functional $W^{-}$ introduced by Montiel and Urbano. We study the energy functional on a family of Hamiltonian-minimal Lagrangian tori and support the Montiel--Urbano conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori and show that if a deformation preserves the conformal type of the torus, then it also preserves the area, i.e. preserves the value of the energy functional. In particular, the deformations generated by Novikov--Veselov equations preserve the area of minimal Lagrangian tori.

We consider closed orientable hypersurfaces in a wide class of warped product manifolds which include space forms, deSitter-Schwarzschild and Reissner-Nordstr\"{o}m manifolds. By using a new integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space. In particular, Corollary 1.8 implies that the embeddedness condition in Theorem 2 of [Brendle-Eichmair, JDG. 2013] is not necessary.

Image retargeting techniques adjust images into different sizes and have attracted much attention recently. Objective quality assessment (OQA) of image retargeting results is often desired to automatically select the best results. Existing OQA methods train a model using some benchmarks (e.g., RetargetMe), in which subjective scores evaluated by users are provided. Observing that it is challenging even for human subjects to give consistent scores for retargeting results of different source images (diff-source-results), in this paper we propose a learning-based OQA method that trains a General Regression Neural Network (GRNN) model based on relative scores --- which preserve the ranking --- of retargeting results of the same source image (same-source-results). In particular, we develop a novel training scheme with provable convergence that learns a common base scalar for same-source-results. With this source specific offset, our computed scores not only preserve the ranking of subjective scores for same-source-results, but also provide a reference to compare the diff-source-results. We train and evaluate our GRNN model using human preference data collected in RetargetMe. We further introduce a subjective benchmark to evaluate the generalizability of different OQA methods. Experimental results demonstrate that our method outperforms ten representative OQA methods in ranking prediction and has better generalizability to different datasets.

Using K-theory, we construct a map \pi from the tangent space to the Hilbert scheme at a point Y to the local cohomology group. We use this map \pi to answer (after slight modification) a question by Mark Green and Phillip Griffiths on constructing a map from the tangent space to the Hilbert scheme at a point to the tangent space to the cycle group.

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