... Let the compact complex manifold M" admit a holomorphic vec-tor field Z. If So</?<... more ... Let the compact complex manifold M" admit a holomorphic vec-tor field Z. If So</?<«i hp,p+i =0 then there exists a sequence of differential forms ceq of type (q, q) such that daq = iz aq +1 for 0 ... It follows that Sizan-.i= —izdoL„-\ = —izan = 0, and so 3a„_2 such that 9ап_2 = iotn-\. ...
Bulletin of the American Mathematical Society, 1977
AMS (MOS) subject classifications (1970). Primary 53C20, 53C30; Secondary 57F15. 1 These results ... more AMS (MOS) subject classifications (1970). Primary 53C20, 53C30; Secondary 57F15. 1 These results form a portion of the author's doctoral dissertation written under the supervision of Professor L. Nirenberg, to whom the author would like to express his gratitude and appreciation.
Proceedings of the American Mathematical Society, 1981
Stokes’ theorem was first extended to noncompact manifolds by Gaffney. This paper presents a vers... more Stokes’ theorem was first extended to noncompact manifolds by Gaffney. This paper presents a version of this theorem that includes Gaffney’s result (and neither covers nor is covered by Yau’s extension of Gaffney’s theorem). Some applications of the main result to the study of subharmonic functions on noncompact manifolds are also given.
Proceedings of the American Mathematical Society, 1985
A general vanishing theorem is proved for elliptic operators. This result is then used to give a ... more A general vanishing theorem is proved for elliptic operators. This result is then used to give a simple proof of the fact that the arithmetic genus vanishes for complex manifolds of odd dimension n n with nowhere zero ( n , 0 ) \left ( {n,0} \right ) form.
For a submanifold M v c> R N , we determine a two-term asymptotic formula for vol (M p f] B E (x)... more For a submanifold M v c> R N , we determine a two-term asymptotic formula for vol (M p f] B E (x)) for xe M v as e j, 0. The second term is a quadratic curvature invariant of the second fundamental form of the imbedding. Imbedded spheres are characterized among compact hypersurfaces by this term.
Let M^ be the n-dimensional real hyperbolic space and tt : IH" M he the universal covering map of... more Let M^ be the n-dimensional real hyperbolic space and tt : IH" M he the universal covering map of a compact Riemannian manifold M of constant curvature-1. Let P C be a fc-dimensional complete totally geodesic submanifold and P^ C M" be the corresponding tubulär hypersurface at distance r. In this article we prove that n(Pr) distributes increasingly uniformly in M as r oo. Using eigenspace decomposition of the Laplacian, this fact can be considered eis geometric application of the asymptotics of a particular ordinary difFerential equation.
Note Added in Proof. (a) The results of Theorem 4.1 are valid even for p> 89 and this result is s... more Note Added in Proof. (a) The results of Theorem 4.1 are valid even for p> 89 and this result is sharp. The additional arguments needed will be described elsewhere (preprint available, 1980). (b) Applications of the results of sections two and three to Brownian motion on noncompact manifolds will also be described elsewhere.
Let M be a noncompact Riemannian manifold with Laplace-Beltrami operator A acting on functions on... more Let M be a noncompact Riemannian manifold with Laplace-Beltrami operator A acting on functions on M, 2=2(M) the bottom of spec (-A), and attendant minimal positive heat kernel p(x,y,t) (where (x,y, t) is an element of M x M x (0, + oo)). In this note we prove the following THEOREM. For all x, y in M we have the existence of the limit lim e;tp(x, y, t) =, .~'(x, y), (1) tT+oo for which we have the following alternative: Either o~ vanishes identically on all of M • M, in which case 2 possesses no L 2 eigenfunctions; or ~ is strictly positive on all of M x M in which case 2 possesses a positive normalized L 2 eigenfunction 49 (normalized in the sense that its L 2 norm is equal to 1)for which lim e;'tp(x, y, t) = 49(x)49(y) (2) tt+oo locally uniformly on all of M • M. Furthermore, if M is noncompact Riemannian complete with bounded geometry (to be explicated below), then lim 49(x) = 0. (3) x~oz~ The simplest example of the case ~-= 0 is [~n, n > 1, in which case we have 2 = O, p(x, y, t) = (4nt)-"/z e-lx-ylZ/4'. ~Supported in part by NSF grant DMS 8704325 and PSC-CUNY FRAP awards. 2Supported in part by NSF grant DMS 8506636 and PSC-CUNY FRAP awards.
Let (M n, ds 2) be an n-dimensional Riemannian manifold, dVthe Riemannian measure, and p(x, y, t)... more Let (M n, ds 2) be an n-dimensional Riemannian manifold, dVthe Riemannian measure, and p(x, y, t) the minimal positive heat kernel, of(M", ds2). Then for any ~0 >_-0 in L~ (M), (Pt~a)(x) = " f p(x,y, t )qJ(y)dV(y) M is the minimal positive solution of the heat equation on (M ~, ds 2) satisfying lim Pt~a = g tlo at all points of continuity of ~0. Thus, for M" --R" with canonical Riemannian metric, we have = f (47tt)-n/2e-Ix-rl2/4t~7(y)dy (P/p)(x) R" where dy is the usual Lebesgue measure. (See [3, Chapters VI-VIII].) Our primary interest is in Riemannian manifolds where for every nonnegative ~o in L~ ~ (M), the locus H(t) = H(t; g)=" ~x " (Pt~)(x)= max (Pt{o)(Y)[ t y J of "hot spots" is compact for all t > 0. The collection of such Riemannian manifolds is reasonably large. Besides the compact ones, the class contains those noncompact Riemannian manifolds for which for all t > 0, r EL~(M), we have (PtqO(x) ~ 0 as x ---oo (see [ 12], [5], [7]
Abstract. This paper is mainly concerned with estimates of spectral gaps of Schrödinger oper-ator... more Abstract. This paper is mainly concerned with estimates of spectral gaps of Schrödinger oper-ators ∆ + q with smooth potential q on real hyperbolic space Hn. The estimates are obtained by explicit constructions of approximate generalized eigenfunctions. Among the results ...
This paper treats various aspects of the asymptotic behavior of solutions of certain elliptic equ... more This paper treats various aspects of the asymptotic behavior of solutions of certain elliptic equations of geometric interest on complete Riemannian manifolds. Sharp results relating the rate of volume growth of a complete Riemannian manifold and the growth of its harmonic and subharmonic functions can be found in E22] together with references to related results. In Sect. 2 of this paper we consider solutions of the more restrictive inequality Au>=e>O (where = div o grad) and show that these must be unbounded if M n has even quadratically exponential volume growth, and that u must have faster than linear growth if t
It should be remarked here that the Euler-Lagrange equations of various (not necessarily regular)... more It should be remarked here that the Euler-Lagrange equations of various (not necessarily regular) multiple integral variational problems are of the form (1.1) where A and B satisfy conditions which take the general form of (1.2) (cf. Serrin 1"21]). In particular, the structural conditions imposed above include, when a = n = 2, the second-order linear elliptic equation in divergence form JOURNAL D'ANALYSE MAT~TIOUE, VoI. 39 (1981)
The main results of this article are small time heat comparison results for two points in two man... more The main results of this article are small time heat comparison results for two points in two manifolds with characteristic functions as initial temperature distributions (Theorems 1 and 2). These results are based on the geometric concepts of (essential) distance from the complement and spherical area function. We also discuss some other geometric results about the heat development and illustrate
... Let the compact complex manifold M" admit a holomorphic vec-tor field Z. If So</?<... more ... Let the compact complex manifold M" admit a holomorphic vec-tor field Z. If So</?<«i hp,p+i =0 then there exists a sequence of differential forms ceq of type (q, q) such that daq = iz aq +1 for 0 ... It follows that Sizan-.i= —izdoL„-\ = —izan = 0, and so 3a„_2 such that 9ап_2 = iotn-\. ...
Bulletin of the American Mathematical Society, 1977
AMS (MOS) subject classifications (1970). Primary 53C20, 53C30; Secondary 57F15. 1 These results ... more AMS (MOS) subject classifications (1970). Primary 53C20, 53C30; Secondary 57F15. 1 These results form a portion of the author's doctoral dissertation written under the supervision of Professor L. Nirenberg, to whom the author would like to express his gratitude and appreciation.
Proceedings of the American Mathematical Society, 1981
Stokes’ theorem was first extended to noncompact manifolds by Gaffney. This paper presents a vers... more Stokes’ theorem was first extended to noncompact manifolds by Gaffney. This paper presents a version of this theorem that includes Gaffney’s result (and neither covers nor is covered by Yau’s extension of Gaffney’s theorem). Some applications of the main result to the study of subharmonic functions on noncompact manifolds are also given.
Proceedings of the American Mathematical Society, 1985
A general vanishing theorem is proved for elliptic operators. This result is then used to give a ... more A general vanishing theorem is proved for elliptic operators. This result is then used to give a simple proof of the fact that the arithmetic genus vanishes for complex manifolds of odd dimension n n with nowhere zero ( n , 0 ) \left ( {n,0} \right ) form.
For a submanifold M v c> R N , we determine a two-term asymptotic formula for vol (M p f] B E (x)... more For a submanifold M v c> R N , we determine a two-term asymptotic formula for vol (M p f] B E (x)) for xe M v as e j, 0. The second term is a quadratic curvature invariant of the second fundamental form of the imbedding. Imbedded spheres are characterized among compact hypersurfaces by this term.
Let M^ be the n-dimensional real hyperbolic space and tt : IH" M he the universal covering map of... more Let M^ be the n-dimensional real hyperbolic space and tt : IH" M he the universal covering map of a compact Riemannian manifold M of constant curvature-1. Let P C be a fc-dimensional complete totally geodesic submanifold and P^ C M" be the corresponding tubulär hypersurface at distance r. In this article we prove that n(Pr) distributes increasingly uniformly in M as r oo. Using eigenspace decomposition of the Laplacian, this fact can be considered eis geometric application of the asymptotics of a particular ordinary difFerential equation.
Note Added in Proof. (a) The results of Theorem 4.1 are valid even for p> 89 and this result is s... more Note Added in Proof. (a) The results of Theorem 4.1 are valid even for p> 89 and this result is sharp. The additional arguments needed will be described elsewhere (preprint available, 1980). (b) Applications of the results of sections two and three to Brownian motion on noncompact manifolds will also be described elsewhere.
Let M be a noncompact Riemannian manifold with Laplace-Beltrami operator A acting on functions on... more Let M be a noncompact Riemannian manifold with Laplace-Beltrami operator A acting on functions on M, 2=2(M) the bottom of spec (-A), and attendant minimal positive heat kernel p(x,y,t) (where (x,y, t) is an element of M x M x (0, + oo)). In this note we prove the following THEOREM. For all x, y in M we have the existence of the limit lim e;tp(x, y, t) =, .~'(x, y), (1) tT+oo for which we have the following alternative: Either o~ vanishes identically on all of M • M, in which case 2 possesses no L 2 eigenfunctions; or ~ is strictly positive on all of M x M in which case 2 possesses a positive normalized L 2 eigenfunction 49 (normalized in the sense that its L 2 norm is equal to 1)for which lim e;'tp(x, y, t) = 49(x)49(y) (2) tt+oo locally uniformly on all of M • M. Furthermore, if M is noncompact Riemannian complete with bounded geometry (to be explicated below), then lim 49(x) = 0. (3) x~oz~ The simplest example of the case ~-= 0 is [~n, n > 1, in which case we have 2 = O, p(x, y, t) = (4nt)-"/z e-lx-ylZ/4'. ~Supported in part by NSF grant DMS 8704325 and PSC-CUNY FRAP awards. 2Supported in part by NSF grant DMS 8506636 and PSC-CUNY FRAP awards.
Let (M n, ds 2) be an n-dimensional Riemannian manifold, dVthe Riemannian measure, and p(x, y, t)... more Let (M n, ds 2) be an n-dimensional Riemannian manifold, dVthe Riemannian measure, and p(x, y, t) the minimal positive heat kernel, of(M", ds2). Then for any ~0 >_-0 in L~ (M), (Pt~a)(x) = " f p(x,y, t )qJ(y)dV(y) M is the minimal positive solution of the heat equation on (M ~, ds 2) satisfying lim Pt~a = g tlo at all points of continuity of ~0. Thus, for M" --R" with canonical Riemannian metric, we have = f (47tt)-n/2e-Ix-rl2/4t~7(y)dy (P/p)(x) R" where dy is the usual Lebesgue measure. (See [3, Chapters VI-VIII].) Our primary interest is in Riemannian manifolds where for every nonnegative ~o in L~ ~ (M), the locus H(t) = H(t; g)=" ~x " (Pt~)(x)= max (Pt{o)(Y)[ t y J of "hot spots" is compact for all t > 0. The collection of such Riemannian manifolds is reasonably large. Besides the compact ones, the class contains those noncompact Riemannian manifolds for which for all t > 0, r EL~(M), we have (PtqO(x) ~ 0 as x ---oo (see [ 12], [5], [7]
Abstract. This paper is mainly concerned with estimates of spectral gaps of Schrödinger oper-ator... more Abstract. This paper is mainly concerned with estimates of spectral gaps of Schrödinger oper-ators ∆ + q with smooth potential q on real hyperbolic space Hn. The estimates are obtained by explicit constructions of approximate generalized eigenfunctions. Among the results ...
This paper treats various aspects of the asymptotic behavior of solutions of certain elliptic equ... more This paper treats various aspects of the asymptotic behavior of solutions of certain elliptic equations of geometric interest on complete Riemannian manifolds. Sharp results relating the rate of volume growth of a complete Riemannian manifold and the growth of its harmonic and subharmonic functions can be found in E22] together with references to related results. In Sect. 2 of this paper we consider solutions of the more restrictive inequality Au>=e>O (where = div o grad) and show that these must be unbounded if M n has even quadratically exponential volume growth, and that u must have faster than linear growth if t
It should be remarked here that the Euler-Lagrange equations of various (not necessarily regular)... more It should be remarked here that the Euler-Lagrange equations of various (not necessarily regular) multiple integral variational problems are of the form (1.1) where A and B satisfy conditions which take the general form of (1.2) (cf. Serrin 1"21]). In particular, the structural conditions imposed above include, when a = n = 2, the second-order linear elliptic equation in divergence form JOURNAL D'ANALYSE MAT~TIOUE, VoI. 39 (1981)
The main results of this article are small time heat comparison results for two points in two man... more The main results of this article are small time heat comparison results for two points in two manifolds with characteristic functions as initial temperature distributions (Theorems 1 and 2). These results are based on the geometric concepts of (essential) distance from the complement and spherical area function. We also discuss some other geometric results about the heat development and illustrate
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