10,354 research outputs found
The Cauchy problem for the homogeneous Monge-Ampere equation, III. Lifespan
We prove several results on the lifespan, regularity, and uniqueness of
solutions of the Cauchy problem for the homogeneous complex and real
Monge-Ampere equations (HCMA/HRMA) under various a priori regularity
conditions. We use methods of characteristics in both the real and complex
settings to bound the lifespan of solutions with prescribed regularity. In the
complex domain, we characterize the C^3 lifespan of the HCMA in terms of
analytic continuation of Hamiltonian mechanics and intersection of complex time
characteristics. We use a conservation law type argument to prove uniqueness of
solutions of the Cauchy problem for the HCMA. We then prove that the Cauchy
problem is ill-posed in C^3, in the sense that there exists a dense set of C^3
Cauchy data for which there exists no C^3 solution even for a short time. In
the real domain we show that the HRMA is equivalent to a Hamilton--Jacobi
equation, and use the equivalence to prove that any differentiable weak
solution is smooth, so that the differentiable lifespan equals the convex
lifespan determined in our previous articles. We further show that the only
obstruction to C^1 solvability is the invertibility of the associated Moser
maps. Thus, a smooth solution of the Cauchy problem for HRMA exists for a
positive but generally finite time and cannot be continued even as a weak C^1
solution afterwards. Finally, we introduce the notion of a "leafwise
subsolution" for the HCMA that generalizes that of a solution, and many of our
aforementioned results are proved for this more general object
The Ricci iteration and its applications
In this Note we introduce and study dynamical systems related to the Ricci
operator on the space of Kahler metrics as discretizations of certain geometric
flows. We pose a conjecture on their convergence towards canonical Kahler
metrics and study the case where the first Chern class is negative, zero or
positive. This construction has several applications in Kahler geometry, among
them an answer to a question of Nadel and a construction of multiplier ideal
sheaves.Comment: v2: shortened introduction. v3: corrected some typos. v4: shortened
to fit in C. R. Acad. Sci. Pari
On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow
In this note we construct Nadel multiplier ideal sheaves using the Ricci flow
on Fano manifolds. This extends a result of Phong, Sesum and Sturm. These
sheaves, like their counterparts constructed by Nadel for the continuity
method, can be used to obtain an existence criterion for Kahler-Einstein
metrics.Comment: v2: 1. added details for the case n=1. 2. added some references. v3:
minor changes. To appear in Transactions of the American Mathematical Societ
On energy functionals, Kahler-Einstein metrics, and the Moser-Trudinger-Onofri neighborhood
We prove that the existence of a Kahler-Einstein metric on a Fano manifold is
equivalent to the properness of the energy functionals defined by Bando, Chen,
Ding, Mabuchi and Tian on the set of Kahler metrics with positive Ricci
curvature. We also prove that these energy functionals are bounded from below
on this set if and only if one of them is. This answers two questions raised by
X.-X. Chen. As an application, we obtain a new proof of the classical
Moser-Trudinger-Onofri inequality on the two-sphere, as well as describe a
canonical enlargement of the space of Kahler potentials on which this
inequality holds on higher-dimensional Fano Kahler-Einstein manifolds.Comment: v2: 1. style and exposition changes made, 2. added Remark 4.5, 3.
added several references. v3: added footnote on p. 8. v4: minor changes. To
appear in Journal of Functional Analysi
Ricci flow and the metric completion of the space of Kahler metrics
We consider the space of Kahler metrics as a Riemannian submanifold of the
space of Riemannian metrics, and study the associated submanifold geometry. In
particular, we show that the intrinsic and extrinsic distance functions are
equivalent. We also determine the metric completion of the space of Kahler
metrics, making contact with recent generalizations of the Calabi-Yau Theorem
due to Dinew, Guedj-Zeriahi, and Kolodziej. As an application, we obtain a new
analytic stability criterion for the existence of a Kahler-Einstein metric on a
Fano manifold in terms of the Ricci flow and the distance function. We also
prove that the Kahler-Ricci flow converges as soon as it converges in the
metric sense
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