From vanishing theorems to estimating theorems: the Bochner technique revisited

Bérard, Pierre

doi:10.1090/s0273-0979-1988-15679-0

Cited by 103 publications

(71 citation statements)

References 28 publications

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“…We prove this proposition using well-known techniques introduced by De Giorgi. Some results similar to the result of this proposition and other methods of proving results of such type can be found in [3, (see also references there). Remark.…”

Section: Introductionsupporting

confidence: 77%

“…To show (3.2) note that by the definition of P2 for any v e P2 such that v\9si = 0, (3)(4)(5)(6) \\F'(uo)v\\L2{U)>(a + b)\\v\\L2{^.…”

Section: Generalizedmentioning

confidence: 99%

“…The inequality (3.14) shows that this mapping is a contraction. Suppose now, that in addition to (3.13) (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) r<--£--, ||z,||<…”

Section: Generalizedmentioning

confidence: 99%

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Number of Solutions with a Norm Bounded by a Given Constant of a Semilinear Elliptic PDE with a Generic Right-Hand Side

Nabutovsky¹

1992

Transactions of the American Mathematical Society

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Abstract.We consider a semilinear boundary value problem -Au+f(u, x) = 0 in ß C S.N and u = 0 on dQ.. We assume that / is a C°°-smooth function and Í2 is a bounded domain with a smooth boundary. For any Csmooth perturbation h(x) of the right-hand side of the equation we consider the function Nh(S) defined as the number of C2+a-smooth solutions u such that ||«||C0(Q) < 5 of the perturbed problem.How "small" JVft(S) can be made by a perturbation h(x) suchthat P||co/q) < e ? We present here an explicit upper bound in terms of e , S and max _||D¿/(ii,x)|| (¿€{0,1,2}). \u\

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Section: Introductionsupporting

confidence: 77%

“…To show (3.2) note that by the definition of P2 for any v e P2 such that v\9si = 0, (3)(4)(5)(6) \\F'(uo)v\\L2{U)>(a + b)\\v\\L2{^.…”

Section: Generalizedmentioning

confidence: 99%

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Number of Solutions with a Norm Bounded by a Given Constant of a Semilinear Elliptic PDE with a Generic Right-Hand Side

Nabutovsky¹

1992

Transactions of the American Mathematical Society

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“…for Yang-Mills connections [13] or other recent works on spectrum estimates for Riemannian manifolds (see the survey [2] for examples). It was noticed by some authors that refined versions of the Kato inequality were also true in some special circumstances.…”

Section: A|^|a||^| (4)mentioning

confidence: 99%

Refined Kato inequalities in riemannian geometry

Herzlich¹

2000

Journées Équations Aux Dérivées Partielles

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“…⊓ ⊔ Remark 11.2.6. For a very nice survey of some beautiful applications of this technique we refer to [10]. ⊓ ⊔ Let (M, J) be an almost complex manifold.…”

Section: The Gauss-bonnet Theorem For Hypersurfaces Of An Euclidean Smentioning

confidence: 99%

Lectures on the Geometry of Manifolds

Nicolaescu

1996

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IntroductionShape is a fascinating and intriguing subject which has stimulated the imagination of many people. It suffices to look around to become curious. Euclid did just that and came up with the first pure creation. Relying on the common experience, he created an abstract world that had a life of its own. As the human knowledge progressed so did the ability of formulating and answering penetrating questions. In particular, mathematicians started wondering whether Euclid's "obvious" absolute postulates were indeed obvious and/or absolute. Scientists realized that Shape and Space are two closely related concepts and asked whether they really look the way our senses tell us. As Felix Klein pointed out in his Erlangen Program, there are many ways of looking at Shape and Space so that various points of view may produce different images. In particular, the most basic issue of "measuring the Shape" cannot have a clear cut answer. This is a book about Shape, Space and some particular ways of studying them.Since its inception, the differential and integral calculus proved to be a very versatile tool in dealing with previously untouchable problems. It did not take long until it found uses in geometry in the hands of the Great Masters. This is the path we want to follow in the present book.In the early days of geometry nobody worried about the natural context in which the methods of calculus "feel at home". There was no need to address this aspect since for the particular problems studied this was a non-issue. As mathematics progressed as a whole the "natural context" mentioned above crystallized in the minds of mathematicians and it was a notion so important that it had to be given a name. The geometric objects which can be studied using the methods of calculus were called smooth manifolds. Special cases of manifolds are the curves and the surfaces and these were quite well understood. B. Riemann was the first to note that the low dimensional ideas of his time were particular aspects of a higher dimensional world.The first chapter of this book introduces the reader to the concept of smooth manifold through abstract definitions and, more importantly, through many we believe relevant examples. In particular, we introduce at this early stage the notion of Lie group. The main geometric and algebraic properties of these objects will be gradually described as we progress with our study of the geometry of manifolds. Besides their obvious usefulness in geometry, the Lie groups are academically very friendly. They provide a marvelous testing ground for abstract results. We have consistently taken advantage of this feature throughout this book. As a bonus, by the end of these lectures the reader will feel comfortable manipulating basic Lie theoretic concepts.To apply the techniques of calculus we need "things to derivate and integrate". These i ii "things" are introduced in Chapter 2. The reason why smooth manifolds have many differentiable objects attached to them is that they can be locally very well approximated by linear sp...

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From vanishing theorems to estimating theorems: the Bochner technique revisited

Cited by 103 publications

References 28 publications

Number of Solutions with a Norm Bounded by a Given Constant of a Semilinear Elliptic PDE with a Generic Right-Hand Side

Number of Solutions with a Norm Bounded by a Given Constant of a Semilinear Elliptic PDE with a Generic Right-Hand Side

Refined Kato inequalities in riemannian geometry

Lectures on the Geometry of Manifolds

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