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174
On the structure of spaces with Ricci curvature bounded below. I
 J. DIFFERENTIAL GEOM
, 1997
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SW ⇒ Gr: From the SeibergWitten equations to pseudoholomorphic curves
 J. Amer. Math. Soc
, 1996
"... The purpose of this article is to explain how pseudoholomorphic curves in a symplectic 4manifold can be constructed from solutions to the SeibergWitten equations. As such, the main theorem proved here (Theorem 1.3) is an existence theorem for pseudoholomorphic curves. This article thus provides ..."
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Cited by 66 (2 self)
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The purpose of this article is to explain how pseudoholomorphic curves in a symplectic 4manifold can be constructed from solutions to the SeibergWitten equations. As such, the main theorem proved here (Theorem 1.3) is an existence theorem for pseudoholomorphic curves. This article thus provides a proof of roughly half of
Geometry and curvature of diffeomorphism groups with H¹ metric and mean hydrodynamics
, 1998
"... In [HMR1], Holm, Marsden, and Ratiu derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equation ˙V (t)+∇U(t)V (t)−α2 [∇U(t)] t ·△U(t) = −grad p(t) where divU = 0, and V = (1 − α2△)U. In this model, the momentum V is transported by the velocity U, with the eff ..."
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Cited by 38 (13 self)
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In [HMR1], Holm, Marsden, and Ratiu derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equation ˙V (t)+∇U(t)V (t)−α2 [∇U(t)] t ·△U(t) = −grad p(t) where divU = 0, and V = (1 − α2△)U. In this model, the momentum V is transported by the velocity U, with the effect that nonlinear interaction between modes corresponding to length scales smaller than α is negligible. We generalize this equation to the setting of an n dimensional compact Riemannian manifold. The resulting equation is the EulerPoincaré equation associated with the geodesic flow of the H1 right invariant metric on Ds µ, the group of volume preserving Hilbert diffeomorphisms of class Hs. We prove that the geodesic spray is continuously differentiable from T Ds µ(M) into TTD s µ(M) so that a standard Picard iteration argument proves existence and uniqueness on a finite time interval. Our goal in this paper is to establish the foundations for Lagrangian stability analysis following Arnold [A]. To do so, we use submanifold geometry, and prove that the weak curvature tensor of the right invariant H1 metric on Ds µ is a bounded trilinear map in the Hs topology, from which it follows that solutions to Jacobi’s equation exist. Using such solutions, we are able to study the infinitesimal stability behavior of geodesics.
Minimizing area among Lagrangian surfaces: the mapping problem
 J. Diff. Geom
, 2000
"... This paper introduces a geometrically constrained variational problem for the area functional. We consider the area restricted to the lagrangian surfaces of a Kähler surface,or,more generally,a symplectic 4manifold with suitable metric,and study its critical points and in particular its minimizers. ..."
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Cited by 34 (6 self)
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This paper introduces a geometrically constrained variational problem for the area functional. We consider the area restricted to the lagrangian surfaces of a Kähler surface,or,more generally,a symplectic 4manifold with suitable metric,and study its critical points and in particular its minimizers. We apply this study to the problem of finding canonical representatives of the lagrangian homology (that part of the homology generated by lagrangian cycles). We show that the lagrangian homology of a Kähler surface (or of a symplectic 4manifold) is generated by minimizing lagrangian surfaces that are branched immersions except at finitely many singular points. We precisely describe the structure of these singular points. In particular,these singular points are represented by lagrangian cones with an associated local Maslov index. Only those cones of Maslov index 1 or −1 may be area minimizing. The mean curvature of the minimizers satisfies a firstorder system of partial differential equations of “Hodgetype”. 1.
Null Lagrangians, weak continuity and variational problems of arbitrary order
 J. FUNCT. ANAL
, 1981
"... We consider the problem of minimizing integral functionals of the form I(u) = jp F(x, ““‘u(x)) dx, where II c IR’, u: $2+ iR4 and V % denotes the set of all partial derivatives of u with orders c/c. ‘The method is based on a characterization of null Lagrangians L(V”u) depending only on derivatives ..."
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Cited by 33 (7 self)
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We consider the problem of minimizing integral functionals of the form I(u) = jp F(x, ““‘u(x)) dx, where II c IR’, u: $2+ iR4 and V % denotes the set of all partial derivatives of u with orders c/c. ‘The method is based on a characterization of null Lagrangians L(V”u) depending only on derivatives of order k. Applications to elasticity and other theories of mechanics are given.
Asymptotically flat initial data with prescribed regularity at infinity
 Comm. Math. Phys
, 2001
"... We prove the existence of a large class of asymptotically flat initial data with nonvanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at spacelike infinity in terms of powers of a radial coordinate. 1 1 ..."
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Cited by 29 (8 self)
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We prove the existence of a large class of asymptotically flat initial data with nonvanishing mass and angular momentum for which the metric and the extrinsic curvature have asymptotic expansions at spacelike infinity in terms of powers of a radial coordinate. 1 1
Approximation of a class of optimal control problems with order of convergence estimates
 J. Math. Anal. Appl
, 1973
"... Submitted by J. L. Lions An approximation scheme for a class of optimal control problems is presented. An order of convergence estimate is then developed for the error in the approximation of both the optimal control and the solution of the control equation. 1. ..."
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Cited by 25 (0 self)
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Submitted by J. L. Lions An approximation scheme for a class of optimal control problems is presented. An order of convergence estimate is then developed for the error in the approximation of both the optimal control and the solution of the control equation. 1.
Extrema Of Curvature Functionals On The Space Of Metrics On 3Manifolds, II.
 Calc. Var. & P.D.E
, 1995
"... This paper is a continuation of the study of some rigidity or nonexistence issues discussed in [An1, x6]. The results obtained here also play a signicant role in the approach to geometrization of 3manifolds discussed in [An4]. Let N be an oriented 3manifold and consider the functional R 2 (g) ..."
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Cited by 22 (14 self)
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This paper is a continuation of the study of some rigidity or nonexistence issues discussed in [An1, x6]. The results obtained here also play a signicant role in the approach to geometrization of 3manifolds discussed in [An4]. Let N be an oriented 3manifold and consider the functional R 2 (g) = Z N jr g j 2 dV g ; (0.1) on the space of metrics M on N where r is the Ricci curvature and dV is the volume form. The EulerLagrange equations for a critical point of R 2 read rR 2 = D Dr +D 2 s 2 R r 1 2 (s jrj 2 ) g = 0; (0.2) s = 1 3 jrj 2 : (0.3) Here s is the scalar curvature, D 2 s the Hessian of s, s = trD 2 s the Laplacian, and R the action of the curvature tensor R on symmetric bilinear forms, c.f. [B, Ch.4H] for further details. The equation (0.3) is just the trace of (0.2). It is obvious from the trace equation (0.3) that there are no nonat R 2 critical metrics, i.e. solutions of (0.2)(0.3), on compact manifolds N ; this follows immediately by integrating (0.3) over N . Equivalently, since the functional R 2 is not scale invariant in dimension 3, there are no critical metrics g with R 2 (g) 6= 0. To obtain nontrivial critical metrics in this case, one needs to modify R 2 so that it is scaleinvariant, i.e. consider v 1=3 R 2 ; where v is the volume of (N; g)
On Leray’s selfsimilar solutions of the NavierStokes equations
 Acta Math
, 1996
"... In the 1934 paper [L] Leray raised the question of the existence of selfsimilar solutions of the Navier{Stokes equations ut u +(u r)u+rp=0 div u =0 ..."
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Cited by 22 (0 self)
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In the 1934 paper [L] Leray raised the question of the existence of selfsimilar solutions of the Navier{Stokes equations ut u +(u r)u+rp=0 div u =0
The Dirichlet Problem for the Total Variation Flow
, 2001
"... We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we u ..."
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Cited by 21 (7 self)
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We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.