Аннотация:
This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be
complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and
diameter bound of the σ2σ2-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.
Ключевые слова:
Bach flat metrics; bubble tree structure; degeneration of metrics; conformally compact; Einstein; renormalized volume.
Поступила:30 августа 2007 г.; в окончательном варианте 7 декабря 2007 г.; опубликована 17 декабря 2007 г.
\RBibitem{ChaQinYan07}
\by Sun-Yung A.~Chang, Jie Qing, Paul Yang
\paper Some Progress in Conformal Geometry
\jour SIGMA
\yr 2007
\vol 3
\papernumber 122
\totalpages 17
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Эта публикация цитируется в следующих 7 статьяx:
Dela Torre A., del Mar Gonzalez M., “Isolated Singularities For a Semilinear Equation For the Fractional Laplacian Arising in Conformal Geometry”, Rev. Mat. Iberoam., 34:4 (2018), 1645–1678
del Mar Gonzalez M., “Recent Progress on the Fractional Laplacian in Conformal Geometry”, Recent Developments in Nonlocal Theory, eds. Palatucci G., Kuusi T., Walter de Gruyter Gmbh, 2018, 236–273
del Mar Gonzalez M., “Renormalized Weighted Volume and Conformal Fractional Laplacians”, Pac. J. Math., 257:2 (2012), 379–394
Leitner F., “Examples of almost Einstein structures on products and in cohomogeneity one”, Differential Geom Appl, 29:3 (2011), 440–462
Rowlett J., “on the Spectral Theory and Dynamics of Asymptotically Hyperbolic Manifolds”, Ann Inst Fourier (Grenoble), 60:7 (2010), 2461–2492
Díaz D.E., “Holographic formula for the determinant of the scattering operator in thermal AdS”, J. Phys. A, 42:36 (2009), 365401, 11 pp.
Weinstein G., Zhang L., “The profile of bubbling solutions of a class of fourth order geometric equations on 4-manifolds”, J. Funct. Anal., 257:12 (2009), 3895–3929
Цитирование в формате AMSBIB
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