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Spectral geometry

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Spectral geometry is a field inmathematics which concerns relationships between geometric structures ofmanifolds andspectra of canonically defineddifferential operators. The case of theLaplace–Beltrami operator on aclosed Riemannian manifold has been most intensively studied, although otherLaplace operators in differential geometry have also been examined. The field concerns itself with two kinds of questions: direct problems and inverse problems.

Inverse problems seek to identify features of the geometry from information about theeigenvalues of the Laplacian. One of the earliest results of this kind was due toHermann Weyl who usedDavid Hilbert's theory ofintegral equation in 1911 to show that the volume of a bounded domain inEuclidean space can be determined from theasymptotic behavior of the eigenvalues for theDirichlet boundary value problem of theLaplace operator. This question is usually expressed as "Can one hear the shape of a drum?", the popular phrase due toMark Kac. A refinement of Weyl's asymptotic formula obtained by Pleijel and Minakshisundaram produces a series of localspectral invariants involvingcovariant differentiations of thecurvature tensor, which can be used to establish spectral rigidity for a special class of manifolds. However as the example given byJohn Milnor tells us, the information of eigenvalues is not enough to determine theisometry class of a manifold (seeisospectral). A general and systematic method due toToshikazu Sunada gave rise to a plethora of such examples which clarifies the phenomenon of isospectral manifolds.

Direct problems attempt to infer the behavior of the eigenvalues of a Riemannian manifold from knowledge of the geometry. The solutions to direct problems are typified by theCheeger inequality which gives a relation between the first positive eigenvalue and anisoperimetric constant (theCheeger constant). Many versions of the inequality have been established since Cheeger's work (byR. Brooks and P. Buser for instance).

References

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Berger, Marcel; Gauduchon, Paul; Mazet, Edmond (1971),Le spectre d'une variété riemannienne, Lecture Notes in Mathematics (in French), vol. 194, Berlin-New York: Springer-Verlag.
Sunada, Toshikazu (1985), "Riemannian coverings and isospectral manifolds",Ann. of Math.,121 (1):169–186,doi:10.2307/1971195,JSTOR 1971195.

Functional analysis (topics –glossary)

Spaces

Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev Topological vector
Properties	Barrelled Complete Dual (Algebraic /Topological) Locally convex Reflexive Separable

Theorems

Operators

Algebras

Open problems

Applications

Advanced topics

Category

v t e Spectral theory and^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With anApproximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

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