Analysts on harmonic maps over geometric singular spaces via Dirichlet forms
| Project/Area Number |
16540201
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Kumamoto University |
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Principal Investigator |
KUWAE Kazuhiro Kumamoto University, Faculty of Education, Associate Professor, 教育学部, 助教授 (80243814)
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| Co-Investigator(Kenkyū-buntansha) |
OGUNA Yukio Kyushu University, Graduate School of Mathematics, Associate Professor, 理工学部, 教授 (00037847)
SHIOYA Takashi Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90235507)
大津 幸男 九州大学, 大学院・数理学研究院, 助教授 (80233170)
MACHIGASHINA Yoshino Osaka Kyoiku University, Faculty of Education, Associate Professor, 教育学部, 助教授 (00253584)
KUWADA Kazumasa Ochanomizu University, Faculty of Science, Lecturer, 理学部, 講師 (30432032)
市田 良輔 横浜市立大学, 国際総合科学部, 教授 (10094294)
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| Project Period (FY) |
2004 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥3,700,000 (Direct Cost: ¥3,700,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2004: ¥1,400,000 (Direct Cost: ¥1,400,000)
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| Keywords | Dirichlet form / harmonic map / subharmonic function / variational convergence / Gromov-Hausdorff convergence / Alexandrov space / Kato class / heat kennel / アレキサンドロフ空間 |
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| Research Abstract |
We establish the following result :
1) Variational convergence of metric measure spaces:
We introduce a natural definition of Lp-convergence of maps. $pge 1$, in the case where the domain is a convergent sequence of measured metric space with respect to the measured Gromov-Hausdorff topology and the target is a Gromov-Hausdorff convergent sequence. With the Lp-convergence, we establish a theory of variational convergences. We prove that the Poincare inequality with some additional condition implies the asymptotic compactness. The asymptotic compactness is equivalent to the Gromov-Hausdorff compactness of the energy-sublevel sets. Supposing that the targets are CAT(0)-spaces, we study convergence of resolvents. As applications, we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of Ricci curvature. This work was done with Prof. T. Shioya.
2) Perturbation of symmetric Markov processes and its related stocha
… More
stic calculus:
We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower orderperturbation of the L2-infinitesimal generator L of a general symmetric Markov process. Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It^o formula for Dirichlet processes is obtained. This work was done with Professors Z.Q. Chen. P.J. Fitzsimmons and T.S. Zhang.
3) Kato class measures over symmetric Markov processes :
We show that $fin L^p(X ; m)$ implies $|f|dmin S_K^1$ for $p>D$ with $D>0$, where $S_K^1$ is a subfamily of Kato class measures relative to a semigroup kennel $p_t(x, y)$ of a Markov process associated with a (non-symmetric) Dirichlet form on $L^2(X ; m)$. We only assume that $p_t(x, y)$ satisfies the Nash type estimate of small time defending on $D$. No concrete expression of $p_t(X, V)$ is needed for the result. This wonk was done with M. Takahashi.
4) Refinements of exceptional sets with respect to (n, p)-capacity oven symmetric Markov processes:
We establish a one to one correspondence between a class of smooth measures in the (n, p)-sense and a class of positive continuous additive functionals admitting (n, p)-exceptional sets. This work was done with A. Sato.
5) Liouville theorems for harmonic maps to convex spaces over Markov chains:
We give a Liouville type theorem for harmonic maps from the space equipped with the harmonicity of functions in terms of conservative Markov chains to convex spaces admitting barycenters. No differentiable structures for the domain and the target are assumed. This work was done with prof. k.Th. Sturm.
6) Laplacian comparison theorem on Alexandrov spaces :
We consider a directionally restricted version of the Bishop-Gromov relative volume comparison as generalized notion of Ricci curvature bounded below for Alexandrov spaces. We prove a Laplacian comparison theorem for Alexandrov spaces under the condition. As an application we prove a topological splitting theorem. This work was done with Prof. T.Shioya. Less
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