[geometry-ml:00788] Workshop on Q-curvature の案内（第２報）

[IZEKI Hiroyasu]

            Workshop on Q-curvature in Conformal Geometry の案内


　約１５年ほど前に，山辺の方程式に現れる共形共変な微分作用素（ラプラシアン
＋スカウテンスカラー）の高階のアナロジーを研究する過程で， "Q-curvature" 
とよばれる不変量が発見されました．
　この "Q-curvature" について，その基礎から最近の話題までを解説していただく
研究集会を，下記の要領で開催いたします．学期末のお忙しい時期だと思いますが，
是非ご参会いただきますよう，ご案内申し上げます．

　日程：２００９年２月１３日（金），１４日（土）
　場所：東北大学理学部数理科学記念館（川井ホール）

　なお，プログラムとアブストラクトは，以下の通りです．

　　　　　　　　　　　　　　　　　　　　　　　組織委員：西川青季，井関裕靖


Program:

February 13, 2009

10:00 - 10:55	Ali Fardoun (Univ ersity of Brest)
                An Introduction to Q-curvature problems, I
11:10 - 12:05	Paul Baird (University of Brest)
                Special features of the Paneitz operator and
                Q-curvature prescription, I
12:20 - 13:15	Ali Fardoun (University of Brest)
                An Introduction to Q-curvature problems, II

15:00 - 15:50	Keisuke Ueno (Yamagata University)
                The Paneitz equation on manifolds with 
		large symmetry
16:00 - 16:50	Hiroyuki Kamada (Miyagi University of Education)
		A class of variational functionals proposed by
		Chang and Fang
17:00 - 17:30	Seiki Nishikawa (Tohoku University)
		A variational problem related to constant
		Q-curvature

February 14, 2009

10:00 - 10:55	Paul Baird (Univ ersity of Brest)
		Special features of the Paneitz operator and
		Q-curvature, II
11:10 - 12:05	Rachid Regbaoui (University of Brest)
		Fully nonlinear equations in conformal geometry, I
12:20 - 13:15	Rachid Regbaoui (University of Brest)
		Fully nonlinear equations in conformal geometry, II


Abstracts:

Ali Fardoun:  An introduction to Q-curvature problems.

    We are interested in the study of "conformally covariant operators",
that is, linear operators which satisfy invariant properties under a
conformal change of the metric on a manifold.  These operators are
related to conformal curvature invariants.  A model example in dimension 
2 is the Laplace operator which is related to the Gaussian curvature.  
    In Lecture 1, we give some basic results and tools for prescribing 
the Gaussian curvature and we discuss how these naturally generalise 
to the Q-curvature introduced by Branson.  The latter curvature is 
naturally associated with the Paneitz operator (which takes on the role 
of the Laplacian in dimension 2) with leading term the bi-Laplace operator.
    In Lecture 2, as for the uniformisation theorem for surfaces, we ask
whether a four-manifold carries a conformal metric whose corresponding 
Q-curvature is constant.  We give sufficient conditions and some 
techniques for the solvability of this problem.

Paul Baird:  Special features of the Paneitz operator and Q-curvature
prescription.

    In dimension 4, the conformally invariant Paneitz operator has many
special properties that make its study both rich and difficult.  As an 
example, unlike the Laplacian, it may possess negative eigenvalues and 
have non-trivial kernel.  Uniformisation theorems are hard to achieve and
often require additional hypotheses.  The classification of manifolds with 
given Q-curvature is not known.
    I will discuss some of these aspects, and in particular, how extremals 
of the associated functional determinant provide a characterisation of 
the round sphere.
    A basic problem is to uniformize the Q-curvature in a given conformal
class.  More generally, given a function on an even dimensional manifold, 
we can try to find a metric in the conformal class with this function as 
its Q-curvature.  In joint work with Ali Fardoun and Rachid Regbaoui, 
we provide two methods to do this.  First, when the Paneitz operator is
positive, we use an evolution equation which seeks to minimize a natural
functional.  In the presence of negative eigenvalues, a minimax method 
is required.

Rachid Regbaoui:  Fully nonlinear equations in conformal geometry

    Recently there has been extensive progress in the study of fully 
nonlinear second order elliptic differential equations (involving second 
order derivatives in the nonlinear terms) on manifolds. This is particulary 
motivated by problems in both classical Euclidean geometry and 
conformal geometry.  
    The type of these equations goes from the Yamabe equation to the
Monge-Amp\{`}ere equations.  Thanks to the work of N.V. Krylov and 
L.C. Evans, there is a general $C^{2+\alpha}$- regularity theory based 
on $C^2$ {\it a priori} estimates.  When dealing with Euclidean domains, 
the work of L. Caffarelli, L. Nirenberg and J. Spruck, gives a good 
understanding of these equations.  In the case of problems arising from 
geometry, the study of these equations depends on the underlying 
geometry.
    In the first lecture, we recall some basic properties of elementary
symmetric functions of symmetric tensors. We then introduce the 
$\sigma_k$-curvature as the $k^{th}$-elementary symmetric function 
of the eigenvalues of the Schouten tensor.  We discuss the differential
equations associated to the prescribed $\sigma_k$-curvature in a fixed 
conformal class.
    In the second lecture, we give a systematic study of the above
equations, and present the recent known results such as existence,
uniquness, regularity, blow up ... etc.
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