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Jiazu Zhou, Fangwei Chen (2007)
THE BONNESEN-TYPE INEQUALITIES IN A PLANE OF CONSTANT CURVATUREJournal of Korean Medical Science, 44
Peter Li (1980)
Eigenvalue estimates on homogeneous manifoldsCommentarii Mathematici Helvetici, 55
L. Santaló (1942)
Integral formulas in Crofton’s style on the sphere and some inequalities referring to spherical curvesDuke Mathematical Journal, 9
S. Yau (1975)
Isoperimetric constants and the first eigenvalue of a compact riemannian manifoldAnnales Scientifiques De L Ecole Normale Superieure, 8
R. Osserman (1978)
The isoperimetric inequalityBulletin of the American Mathematical Society, 84
Xingxing Wang, Wenxue Xu, Jiazu Zhou (2017)
Some logarithmic Minkowski inequalities for nonsymmetric convex bodiesScience China Mathematics, 60
J. Cheeger (1968)
The relation between the laplacian and the diameter for manifolds of non-negative curvatureArchiv der Mathematik, 19
Hung-hsi Wu (1991)
The Estimate of the First Eigenvalue of a Compact Riemannian Manifold
W Blaschke (1956)
Kreis und Kugal
J Cheeger (1970)
Problems in Analysis: A Symposium in Honor of Salomon Bocher
J Zhou (2007)
Bonnesen-type inequalities on the planActa Math. Sin., 50
R Osserman (1978)
Proceedings of the International Congress of Mathematicians
S Cheng (1975)
Eigenfunctions and eigenvalues of LaplacianProc. Symp. Pure Math., 27
R. Osserman (2010)
Isoperimetric Inequalities and Eigenvalues of the Laplacian Robert Osserman
(1961)
Caractérisation variationnelle d’une somme de valeurs propres consécutives
Miao Luo, Wenxue Xu, Jiazu Zhou (2015)
Translative containment measure and symmetric mixed isohomothetic inequalitiesScience China Mathematics, 58
Daniel Klain (2007)
Bonnesen-type inequalities for surfaces of constant curvatureAdv. Appl. Math., 39
J Zhou, F Chen (2007)
The Bonnesen-type inequalities in a plane of constant curvatureJ. Korean Math. Soc., 44
S Yau (1975)
Isoperimetric constants and the first eigenvalue of a compact Riemannian manifoldAnn. Sci. Éc. Norm. Supér., 8
C. Zeng, Lei Ma, Jiazu Zhou, Fangwei Chen (2012)
The Bonnesen isoperimetric inequality in a surface of constant curvatureScience China Mathematics, 55
R. Osserman (1979)
BONNESEN-STYLE ISOPERIMETRIC INEQUALITIESAmerican Mathematical Monthly, 86
I. Chavel, E. Feldman (1974)
The first eigenvalue of the laplacian on manifolds of non-negative curvatureCompositio Mathematica, 29
Q. Cheng, Xuerong Qi (2011)
INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN
J. Cheeger (1969)
A lower bound for the smallest eigenvalue of the Laplacian
Shiu-yuen Cheng (1975)
Eigenvalue comparison theorems and its geometric applicationsMathematische Zeitschrift, 143
R. Schoen, S. Yau (1994)
Lectures on Differential Geometry
(1956)
Kreis und Kugal, 2nd edn
Zengle Zhang, Jiazu Zhou (2017)
Bonnesen-style Wulff isoperimetric inequalityJournal of Inequalities and Applications, 2017
Lluís Santaló (1976)
Integral geometry and geometric probability
Mu-Fa Chen, Feng-Yu Wang (1997)
General formula for lower bound of the first eigenvalue on Riemannian manifoldsScience in China Series A: Mathematics, 40
R Osserman (1978)
The isoperimetric inequalitiesBull. Am. Math. Soc., 84
Peter Li, S. Yau (1980)
Estimates of eigenvalues of a compact Riemannian manifold
(2009)
Modern differential geometry: spectral theory and isospectrum problems, curvature and topological invariants
School of Mathematics and By Cheeger’s isoperimetric constants, some lower bounds and upper bounds of λ , Statistics, Southwest University, Chongqing, 400715, China the first eigenvalue on a complete surface of constant curvature, are given. Some College of Science, Wuhan Bonnesen-style inequalities and reverse Bonnesen-style inequalities for the first University of Science and eigenvalue are obtained. Those Bonnesen-style inequalities obtained are stronger Technology, Wuhan, Hubei 563006, China than the well-known Osserman’s results and the upper bound is stronger than Osserman’s results (Osserman in Proceedings of the International Congress of Mathematicians, Helsinki, 1978). MSC: 53A25; 53A10; 53C23 Keywords: the first eigenvalue; Cheeger’s isoperimetric constants; Bonnesen-style inequality 1 Introduction The classical isoperimetric problem is to determine a plane figure of largest possible area whose boundary has a specific length and it was known in Ancient Greece. However, the first mathematically rigorous proof was obtained only in the nineteenth century and it was well recognized by Weierstrass though Bernoulli, Euler and Lagrange once claimed the proof that was found flawed later. Hurwitz published a short proof using the Fourier series that applies to arbitrary plane domain D whose boundary ∂D was not assumed to be smooth. An elegant direct proof, based on the comparison of a smooth simple closed curve with a circle, was given by Schmidt in by using only the arc length formula, expression for the area of a plane region from Green’s theorem, and the Cauchy-Schwarz inequality []. Many other proofs have been found and some of them were stunningly simple. The isoperimetric problem has been extended in multiple ways, for example, to domains on surfaces and in higher dimensional spaces, or more generally to integral cur- rents and analytic manifolds, but the proof is too difficult. Let D be a domain (subset with nonempty interiors) in the Euclidean plane R with the boundary composing of the simple curve of length L and area A.Then L –πA ≥ , (.) the equality holds when and only when D is a disc. © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Fang and Zhou Journal of Inequalities and Applications (2017) 2017:190 Page 2 of 11 It is known that the isoperimetric inequality (.) is equivalent to the following Sobolev inequality (see []): If f has compact support in D ⊂ R ,then |∇f | –π f ≥ . (.) D D Here ∇ denotes the gradient operator. The equality holds in (.)ifand only if f is the characteristic function of balls. During s, Bonnesen proved a series of inequalities of the form L –πA ≥ B,(.) where the quantity B on the right-hand side is a non-negative geometric invariant of significance and vanishes only when D is a disc. An inequality of the form (.) is called the Bonnesen-style inequality, and it is stronger than the classical isoperimetric inequality. The Bonnesen-style inequality has been ex- tended to surfaces of constant curvature and higher dimensions and many Bonnesen- style inequalities have been found during the past. Mathematicians are still working on unknown Bonnesen-style inequalities of geometric significance [–]. The isoperimetric inequality for domains on surfaces M of constant curvature can be stated as follows. Let D be a compact domain on the surface M of constant curvature. Let A and L denote the area and the boundary length of D,respectively. Then L –πA + MA ≥ , the equality holds if and only if D is a geodesic disc. The Bonnesen-style inequality for domains on surfaces of constant curvature was first investigated by Santaló [, ]. Klain obtained some new Bonnesen-style inequalities for domains on surfaces of constant curvature []. By the kinematic formulas in integral ge- ometry, Xu, Zhou et al. also obtained Bonnesen-style inequalities on a complete surface of constant curvature (see [, ]). Osserman [] studied the Bonnesen-style inequality for the domains on surfaces with the bounded Gauss curvature. More Bonnesen-style ho- mothetic (Wulff) inequalities were obtained in [, , ]. Another important extension of the isoperimetric problem in analysis is eigenvalues of the Laplacian. Eigenvalues of the Laplacian.Let D be a domain with smooth boundary ∂D on a compact Riemannian surface M. The eigenvalue problem u + λu = in D; u| =, ∂D is known to have a complete system of eigenfunctions u = φ , with corresponding eigen- values λ ,where = λ < λ < λ < ··· < λ < ··· ∞. p Fang and Zhou Journal of Inequalities and Applications (2017) 2017:190 Page 3 of 11 One would ask naturally a basic question: how are the properties of domain D on a compact Riemannian surface M, that is, area of D ⊂ M,lengthand integralsofcurvature of ∂D, reflected in the set of eigenvalues {λ }? In this paper, we will investigate the Bonnesen-style inequalities for the first eigenvalue λ of Laplacian on the complete surface. Let M be a compact Riemannian surface and be the Laplacian-Beltrami operator acting on differential functions C (M). It is known that is an elliptic operator. The first eigenvalue λ on domain D ⊂ M can be also char- acterized by [] |∇f | λ (D)= inf,(.) f ∈F |f | where F is the set of smooth functions in D vanishing on the boundary. The Laplace operator on a Riemannian manifold, its spectral theory and the relations between its first eigenvalue and the geometrical data of the manifold, such as curvatures, diameter, injectivity radius and volume, have been extensively studied in the recent math- ematical literature. Amazing connections between the isoperimetric inequality and the first eigenvalue of Laplacian operator have been found during the past decades. One of the basic results is that Cheeger connected the first eigenvalue λ of the Laplacian on a manifold to certain isoperimetric constants. For a domain D on a two-dimensional sur- face, Cheeger considers the quantity h = inf,(.) D ⊂F A where F is the family of relatively compact subdomains of D, A and L are the area and the boundary length of subdomain D ⊂ D, respectively. Cheeger proved that λ (D) ≥ h.(.) The upper estimate of the first eigenvalue of Laplacian has been discussed by geometers and analysts. Hersch [] obtained an upper bound for manifolds homeomorphic to the two sphere. Cheeger [], Chavel and Feldman [] obtained upper bound for manifolds with non-negative Ricci curvature. The comparison theorem of Cheng []gives asharp upper bound for general Riemannian manifold in terms of the Ricci curvature and the diameter of domain. While the progress has been made on the upper bound, not too much is known about the lower bound of the first eigenvalue. The best result is due to Lichnerowicz []who gives a computable sharp lower bound for manifolds whose Ricci curvature is bounded from below by a positive constant. Cheeger [] also gives a lower estimate for general manifolds in terms of some isoperimetric constants. These constants of Cheeger, however, are not computable. Cheng [] observed that if the manifold is a two-dimensional convex surface, then the isoperimetric constant has a lower bound in terms of the diameter. Since , Li and Yau have been trying to obtain the lower bound of the first eigenvalue [, ]. Chen introduced the method in probability theory to improve almost all results proved by others in []. For more detailed isoperimetric properties and the first eigenvalue, one can referto[, , –]. Fang and Zhou Journal of Inequalities and Applications (2017) 2017:190 Page 4 of 11 In [], Osserman considered the first eigenvalue λ on the two-dimensional manifolds with bounded Gauss curvature and obtained some lower and upper bounds of the first eigenvalue by using Cheeger’s isoperimetric constant as follows: Let S be a compact simply connected surface with Gauss curvature K, K ≤ –α , α >. If D ⊂ S is simply connected and ρ is inradius of D,then α α λ (D)– ≥ csch αρ.(.) In this paper, we obtain the following lower bound of the first eigenvalue that is stronger than Osserman’s result (.): Let S be a simply connected complete surface with Gauss curvature K ≤ everywhere. For any simply connected domain D ⊂ S, let ρ denote itsinradiusand R denoteits circum- radius. Let α = inf(–K), β = sup(–K), < α ≤ β, then α α λ (D)– ≥ csch αρ + B, where the quantity B is a positive number depending on α, β, ρ,R. We also obtain the upper bound of the first eigenvalue. By Cheng’s eigenvalue compari- son theorem ([], Theorem .), we obtain a stronger upper bound of the first eigenvalue λ (Theorem .). 2 The Bonnesen-style isoperimetric inequalities Let D denote the geodesic disc of radius r on the complete simply connected surface of M M constant Gauss curvature K ≡ M.Let A , L be, respectively, the area and the length of r r boundary of D . Then the explicit expressions for these quantities are αr sinh sinh αr cosh αr – M M M =–α <: L =π , A =π =π ; r r α α α M M M =: L =πr, A = πr ; r r αr sin sin αr – cos αr M M M = α >: L =π , A =π =π . r r α α α For the geodesic disc, the following equation can be easily verified in all three cases: M M M L –πA + M A =. (.) r r r The isoperimetric inequality on a surface of constant curvature K ≡ M is L –πA + MA ≥ . (.) Namely, given a domain D of area A,if r is chosen so that A equals A,then(.)and (.) M M imply L ≥ L , so that the disc D has minimum boundary length among all domains of r r the same area. Fang and Zhou Journal of Inequalities and Applications (2017) 2017:190 Page 5 of 11 Osserman considered the isoperimetric inequality of two-dimensional complete surface with bounded Gauss curvature []. Let D be a simply connected domain whose Gauss curvature K satisfies K ≤ M.Let L and A be the boundary length and the area of D,respectively. Then L –πA + MA ≥ , (.) where the equality holds if and only if K ≡ M and D is a geodesic disc. Osserman also obtained the following Bonnesen-style isoperimetric inequalities. Theorem A ([]) Let D be a simply connected domain whose Gauss curvature K satisfies K ≤ M. Let ρ be the inradius of D, A be the area of D and L be the length of its boundary. Then the following inequalities are equivalent: M M M LL + MAA ≥ π A + A,(.) ρ ρ ρ M L –πA + MA ≥ L – A,(.) M M L –πA + MA ≥ L – L + M A – A.(.) ρ ρ Moreover, if MA <π, then these inequalities are equivalent to π M L –πA + MA ≥ A – A.(.) Osserman estimated lower bounds of the first eigenvalue by Cheeger’s isoperimetric constants as follows. Theorem B ([]) Let S be a simply connected complete surface with Gauss curvature K, K ≤ –α , α >. Then, for any domain D ⊂ S of circumradius R, α α λ (D)– ≥ (csch αR).(.) If D is simply connected and ρ is its inradius, then α α λ (D)– ≥ (csch αρ).(.) 3 The lower bound of λ In this section, we give some lower bounds of the first eigenvalue λ by Cheeger’s isoperi- metric constants and Bonnesen-style isoperimetric inequalities. We need the following lemmas. Lemma . Let f (r) be continuously differentiable on the interval ≤ r ≤ r . Suppose that, except at a finite number of the points in the interval, f (r) exists and satisfies f (r)+ cf (r) ≤ , f () = , f () = a (.) Fang and Zhou Journal of Inequalities and Applications (2017) 2017:190 Page 6 of 11 for some constants a, c. Let h(r) be the unique solution of h (r)+ ch(r)=, h() = , h () = . (.) Let s be any number such that h(r)> for < r < s, and let r = min{r , s}. Then f (r) ≤ ah(r) (.) for ≤ r ≤ r . f (r) Proof Let φ(r)= .Thenby(.)and (.) h(r) h φ = f h – fh = f h – fh ≤ , except at the singular points. By the mean value theorem, h φ is a weakly monotone de- creasing function, and hence h φ (r) ≤ h φ () = , ≤ r ≤ r . That is, φ (r) ≤ , and hence f (r) f (r) = φ(r) ≤ φ() = lim = a h(r) r→ h(r) for ≤ r ≤ r . Lemma . Let D be a geodesic disc of radius ρ, and let A be the area of D . If M ≤ K ≤ ρ ρ ρ on D , then A ≤ A,(.) where equality holds if and only if K ≡ Mon D . Proof We introduce geodesic polar coordinates in D .The metric canbewritten as ds = dr + g(r, θ) dθ ,where foreach θ, the function f (r)= g(r, θ)satisfies f () = , f () = . √ ∂ Since K ≤ , the geodesic disc of radius ρ always exists. Then with the fact K =– √ g g ∂r and the condition M ≤ K ≤ , f (r)satisfies (.), with a =, c = M, r = ρ.By(.), we have π ρ ρ A = g(r, θ) dr dθ ≤ π h(r) dr = A.(.) Since h(r) can be written explicitly as h(r)= L ,itsatisfies(.)and A = L(r) dr. ρ π The equality holds if and only if g(r, θ) ≡ h(r), hence K ≡ M. Theorem . Let S be a simply connected complete surface with Gauss curvature K ≤ everywhere. For any simply connected domain D ⊂ S, let A, L, Rbe the area, the boundary length and the circumradius of D, respectively. Let α = inf(–K), β = sup(–K), < α ≤ β,(.) D Fang and Zhou Journal of Inequalities and Applications (2017) 2017:190 Page 7 of 11 then α β βR λ (D)– ≥ csch.(.) Proof By Lemma . and (.), we have –β A ≥ A ≥ A.(.) Via (.), the isoperimetric inequality (.)can be rewrittenas L ≥ π + α . A A Then, by (.), we have L ≥ π + α A A ≥ π + α –β βR = β csch + α.(.) By (.)and (.), then α β βR λ (D)– ≥ csch . We complete the proof. Since is monotonically decreasing for x ≥ , hence (.)isstrongerthan(.)if sinh x β < α.By(.) we obtain a lower bound of λ that is stronger than the one in (.). Theorem . Let S be a simply connected complete surface with Gauss curvature K ≤ everywhere. For any simply connected domain D ⊂ S, let A, L, ρ, Rbethearea, the boundary length, the inradius and the circumradius of D, respectively. Let α = inf(–K), β = sup(–K), < α ≤ β,(.) then α α λ (D)– ≥ (csch αρ) + B,(.) where αρ αρ β sinh β sinh α α coth αρ B = + . βR βR sinh αρ sinh αρ α sinh α sinh Fang and Zhou Journal of Inequalities and Applications (2017) 2017:190 Page 8 of 11 Proof By (.)and (.), we have –α –α –α L ≥ π + + α A , ρ ρ A A and hence, –α L α ≥ + α coth αρ.(.) A A sinh αρ By (.)and (.), (.)can be rewrittenas –α L α ≥ + α coth αρ A A sinh αρ –α ≥ + α coth αρ –β sinh αρ αρ β sinh = + α coth αρ. βR sinh αρ α sinh By (.)and (.), we have αρ β sinh α λ (D) ≥ + α coth αρ βR sinh αρ α sinh = (coth αρ) + B α α = + (csch αρ) + B, and αρ αρ β sinh α β sinh α coth αρ B = + . βR βR sinh αρ sinh αρ α sinh α sinh Since B ≥ , hence inequality (.) is stronger than inequality (.). Let R = ρ in Theo- rem .,thatis, let D be a geodesic disc with radius ρ on S.Thenlet ρ →∞ in (.), we immediately obtain the following. Corollary . Let S be a simply connected complete surface with K ≤ –α , α > every- where, then λ (S) ≥ . Next, we give a lower bound of the first eigenvalue λ . Theorem . Let D be a complete simply connected surface and A denote the area of D. Then π λ (D) ≥ . A Fang and Zhou Journal of Inequalities and Applications (2017) 2017:190 Page 9 of 11 Proof By Sobolev inequality (.) and Hölder’s inequality, we have π f dx ≤ |∇f | dx D D ≤ dx |∇f | dx D D = A |∇f | dx, that is, |∇f | dx π ≥ . f dx A By (.), we have |∇f | dx π λ = inf ≥ . f = f dx A By Lemma ., we obtain a lower bound of λ . Corollary . Let S be a simply connected complete surface with Gauss curvature K, –β ≤ K ≤ everywhere. Suppose D ⊂ S, then βR λ (D) ≥ β csch . Here R is the circumradius of D. 4 The upper bound of λ In this section, we consider the upper bound of the first eigenvalue λ .Westart with the following eigenvalue comparison theorem proved by Cheng in []. Denote the open geodesic ball of radius r with center x by D(x; r). Denote by V (M; r) the geodesic ball of radius r in the n-dimensional simply connected space form with constant sectional cur- vature M.Wewrite λ (D(x; r)) as λ (D(x; r)). Theorem C Suppose that S is a complete Riemannian manifold and Ricci curvature of S ≥ (n –)M, n = dim S. Then, for x ∈ S, we have λ D(x; r) ≤ λ V (M; r) n and equality holds if and only if D(x; r) is isometric to V (M; r). In particular, the eigenvalue comparison theorem is also valid when S is a two- dimensional complete simply connected surface. Corollary . Suppose that S is a complete simply connected surface with Gauss curvature K ≥ M. Let D be a geodesic disc with radius r on S, then λ (D ) ≤ λ D , r where equality holds if and only if K ≡ Mon S. Fang and Zhou Journal of Inequalities and Applications (2017) 2017:190 Page 10 of 11 The next lemma will be needed in proving our theorem. Lemma . ([]) Suppose that S is a simply connected complete surface with Gauss cur- vature K ≤ everywhere. Let D be a geodesic disc of radius ρ. If α = inf(–K), β = sup(–K), < α ≤ β,(.) then β π λ (D ) ≤ +.(.) ρ α coth αρ ρ Combining Corollary . and Lemma . immediately yields the following. Theorem . Suppose that S is a simply connected complete surface with Gauss curvature K,–β ≤ K ≤ everywhere. Let D be a geodesic disc of radius ρ, then β π λ (D ) ≤ +.(.) ρ coth βρ ρ –β Proof Since D satisfies the hypotheses of Lemma . when α = β,hence β π –β λ D ≤ + . coth βρ ρ By Corollary ., we immediately obtain (.). We complete the proof of Theorem .. Since the function x coth x is monotonically increasing for x ≥ , hence inequality (.) is stronger than (.). Let ρ →∞ in (.), we can easily have the following corollary. Corollary . Let S be a simply connected complete surface with Gauss curvature K, –β ≤ K ≤ everywhere. Then λ (S) ≤ . Acknowledgements The authors would like to thank anonymous referees for helpful comments and suggestions that directly led to the improvement of the original manuscript. This paper is supported in part by Natural Science Foundation Project (grant number: #11671325). Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors read and approved the final manuscript. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 1 July 2017 Accepted: 2 August 2017 Fang and Zhou Journal of Inequalities and Applications (2017) 2017:190 Page 11 of 11 References 1. Osserman, R: Isoperimetric inequalities and eigenvalues of the Laplacian. In: Proceedings of the International Congress of Mathematicians, Helsinki (1978) 2. Blaschke, W: Kreis und Kugal, 2nd edn. de Gruyter, Berlin (1956) 3. Osserman, R: The isoperimetric inequalities. Bull. Am. Math. Soc. 84(6), 1182-1238 (1978) 4. Luo, M, Xu, W, Zhou, J: Translative containment measure and symmetric mixed isohomothetic inequalities. Sci. China Math. 58, 2593-2610 (2015) 5. Osserman, R: Bonnesen-style isoperimetric inequalities. Am. Math. Mon. 86, 1-29 (1978) 6. Zhou, J: Bonnesen-type inequalities on the plan. Acta Math. Sin. 50, 1397-1402 (2007) 7. 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Journal of Inequalities and Applications – Springer Journals
Published: Aug 15, 2017
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