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Preprints |
C. Arezzo, F. Pacard and M. Singer. Extremal Kähler metrics on blow-ups.
Abstract
: We present some new results about the existence of extremal
Kähler metrics on blow-ups of Kähler manifolds. As a
consequence of our results, we obtain extremal metrics on the blow up
of P^m at n=1, ... , m+1 linearly independent points.
Here is
the final version 
and also the original version
M. Musso, F. Pacard and J. Wei. Finite energy, sign changing solutions with dihedral symmetry for the stationary non linear Schrödinger equation.
Abstract
:
We address the problem of the existence of finite energy solitary
waves for nonlinear Klein-Gordon or Schrodinger type equations. Under
natural conditions on the nonlinearity, we prove the existence of
infinitely many nonradial solutions in any dimension N ≥ 2. Our
result complements earlier works of Bartsch and Willem and Lorca-Ubilla
where solutions invariant under the action of O(2) × O(N −
2) are constructed. In contrast, the solutions we construct are
invariant under the action of Dk × O(N − 2) where Dk ⊂
O(2) denotes the subgroup generated by the rotation of angle 2π/k,
for some integer k ≥ 7, but they are not invariant under the
action of O(2) × O(N − 2).
F. Helein, L. Hauswirth and F. Pacard. A note on some overdetermined problem.
Abstract
:
In this short note, we address the classification of all flat surfaces M
with smooth boundary on which there exist positive harmonic
functions having 0 Dirichlet data and constant (nonzero) Neumann data.
In particular, we show that this problem bear strong similarities with
the study os minimal surfaces in Euclidean 3-space. We also provide a
Weierstrass type representation formula for these surfaces.
M. del Pino, M. Musso, F. Pacard and A. Pistoia. Torus action on S^n and sign changing solutions for conformally invariant equations.
Abstract
:
We construct sequences of sign changing solutions for some conformally
invariant semilinear elliptic equation which is defined in S^n, with n ≥
4. The solutions we obtain have large energy and concentrate along some
special submanifolds of S^n. For example, in dimension n ≥
4 we obtain sequences of solutions whose energy concentrates along one
great circle or finitely many great circles which are linked (a Hopf
link). In dimension n ≥ 5, we obtain sequences of solutions whose energy concentrates along a two dimensional torus (a Clifford torus).
F. Pacard. Constant scalar curvature and extremal metrics on blow ups.
Abstract
:
In this paper, we report some joint works with C. Arezzo and M. Singer
concerning the construction of extremal Kahler metrics on blow ups at
finitely many points of Kahler manifolds which already carry an
extremal metric. In particular, we give sufficient conditions on the
number and locations of the blow up points for the blow up to carry an
extremal Kahler metrics.
F. Pacard. Lectures on "Connected sum
constructions in geometry and nonlinear analysis".
Abstract
: These are notes which cover the lectures I gave in the spring
of 2006 in Roma I-La Sapienza and they also cover the lectures I gave
in the spring of 2007 in the ETH for some Nachdiplomvorlesung. These lectures cover
the analysis of some class of elliptic second order operators on
manifolds with asymptotically periodic ends which appear in the
analysis of some geometric problems : constant mean curvature surfaces,
singular Yamabe problem, ...
So far the first part is complete. It contains all the necessary
information to understand the linear analysis for some class of
ellitpic operators which are defined on manifolds with asymptotically
periodic ends.
The second part will contain applications of the results of the
first part for the understanding of the theory of constant mean
curvature surfaces, complete constant scalar curvature metrics and also some
application to some singular perturbation problems. This part is not yet finished but I am working
on it...and I will do my best to update the latest version on this web page.
