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Dualities in comparison theorems and bundle-valued generalized harmonic forms on noncompact manifolds

Science China Mathematics 2021年第7期64卷 1649-1702页

作者： Shihshu Walter Wei Department of Mathematics University of OklahomaNormanOK 73019USA

We observe,utilize dualities in differential equations and differential inequalities(see Theorem 2.1),dualities between comparison theorems in differential equations(see Theorems E and 2.2),and obtain dualities in"swapping"comparison theorems in differential *** dualities generate comparison theorems on differential equations of mixed typesⅠandⅡ(see Theorems 2.3 and 2.4)and lead to comparison theorems in Riemannian geometry(see Theorems 2.5 and 2.8)with analytic,geometric,PDE's and physical *** particular,we prove Hessian comparison theorems(see Theorems 3.1-3.5)and Laplacian comparison theorems(see Theorems 2.6,2.7 and 3.1-3.5)under varied radial Ricci curvature,radial curvature,Ricci curvature and sectional curvature assumptions,generalizing and extending the work of Han-Li-Ren-Wei(2014)and Wei(2016).We also extend the notion of function or differential form growth to bundle-valued differential form growth of various types and discuss their interrelationship(see Theorem 5.4).These provide tools in extending the notion,integrability and decomposition of generalized harmonic forms to those of bundle-valued generalized harmonic forms,introducing Condition W for bundle-valued differential forms,and proving the duality theorem and the unity theorem,generalizing the work of Andreotti and Vesentini(1965)and Wei(2020).We then apply Hessian and Laplacian comparison theorems to obtain comparison theorems in mean curvature,generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds,the embedding theorem for weighted Sobolev spaces of functions on manifolds,geometric differential-integral inequalities,generalized sharp Hardy type inequalities on Riemannian manifolds,monotonicity formulas and vanishing theorems for differential forms of degree k with values in vector bundles,such as F-Yang-Mills fields(when F is the identity map,they are Yang-Mills fields),generalized Yang-Mills-Born-Infeld fields on manifolds,Liouville type th

关键词： radial curvature Hessian Laplacian Caffarelli-Kohn-Nirenberg inequality F-harmonic map F-Yang-Mills field

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Smooth Line Drawing on Graphics Hardware

Smooth Line Drawing on Graphics Hardware

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第七届中国计算机图形学大会

作者： Hanli Zhao1 , Xiaogang Jin1 , Jianbing Shen2 , Feifei Wei1 , Jieqing Feng1 1 (State Key lab of CAD & CG, Zhejiang University, Hangzhou 310058, China) 2(Indiana University-Purdue University Indianapolis, USA)

This paper presents a new object-space based algorithm for rendering a 3D model as a line drawing using graphics hardware, based on the insight that a line drawing can effectively convey shape using remarkably minimal visual content. Suggestive contours, a new type of line can be combined with silhouettes to produce impressive view-dependent line drawing of smooth shapes. In addition, geometry shader, which is newly introduced into the Direct3D 10 pipeline, can process per-triangle primitive data and output zero or more line primitives. We propose a parallelized line drawing approach that directly extracts 3D sparse linear features in geometry shader. The method selects lines at an appropriate scale automatically, and exhibits good frame-to-frame coherence at interactive rates. The implementation of our algorithm is straightforward, and several experimental examples are given at the end of the paper to demonstrate the effectiveness of our approach.

关键词： Line drawing silhouette suggestive contour radial curvature graphics hardware.

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