Shar gradient estimate and sectral rigidity for -Lalacian Chiung-Jue Anna Sung and Jiaing Wang To aear in ath. Research Letters. Abstract We derive a shar gradient estimate for ositive eigenfunctions of the -Lalacian on a comlete manifold with Ricci curvature bounded below. As an alication, we study the rigidity of manifolds achieving the maximum value of the rincial eigenvalue of the -Lalacian. 1 Introduction In this aer, we consider the -Lalacian L, 1 < <, on a Riemannian manifold given by L(v) = div( v v). A function v is called -harmonic if div( v v) = 0. For a ositive -harmonic function v, if u = ( 1) ln v, then it satisfies div( u u) = u. As aroaches 1, the equation formally reduces to ( ) u div = u. u Partially suorted by NSC and NCTS. Partially suorted by NSF grant No. DS-1105799 1
According to [H-I], the level set of a roer solution to this last equation yields a weak solution to the inverse mean curvature flow. R. oser [] has successfully exloited this oint of view and established the existence of a weak solution to the inverse mean curvature flow starting from the boundary of any smooth comact domain in the Euclidean sace. The crucial oint is to justify the limiting rocess of aroaching 1 as alluded above. This relies on a uniform gradient estimate, indeendent of, for function u = ( 1) ln v with v being -harmonic. Kotschwar and Ni [K-N] have also succeeded in carrying out the same scheme on general Riemannian manifolds with sectional curvature bounded from below. Again, they roceed by obtaining such a uniform gradient estimate. In fact, the estimate itself only involves the lower bound of the Ricci curvature of the underlying manifold. In view of this, it is natural to wonder if the assumtion of the sectional curvature being bounded from below is necessary. In an attemt to address this question, Wang and Zhang [W-Z] have introduced a new aroach for such a gradient estimate. Assuming only the Ricci curvature is bounded from below, they were able to obtain a gradient estimate for u. However, their estimate does not seem to be uniform in. One of the uroses here is to comletely answer the question by roviding a shar gradient estimate for u. In fact, we consider more generally ositive eigenfunctions of the -Lalacian. Theorem 1.1. Let ( n, g) be an n-dimensional comlete noncomact manifold with Ric (n 1). Then for a ositive function v satisfying div( v v) = λv 1, we have ln v y, where y is the unique ositive root of the equation ( 1) y (n 1) y 1 + λ = 0. In articular, we have the following corollary for ositive -harmonic functions. Corollary 1.. Let ( n, g) be an n-dimensional comlete noncomact manifold with Ric (n 1). Let v be a ositive -harmonic function and u = ( 1) ln v. Then u n 1. The estimate in the theorem is shar as demonstrated by the following examle.
Examle 1.3. Let n = R N n 1 with the wared roduct metric ds = dt + e t ds N, where N is a comlete manifold with nonnegative Ricci curvature. Then it can be directly checked (see [L]) that Ric (n 1). Now consider the function v(t, z) = e αt on with n 1 α n 1, where 1 t R and z N. A straightforward calculation shows Hence, L(v) = div( v v) = ( ( 1) α (n 1) ) α 1 v 1. and ln v = α. Clearly, λ = ( n 1 ( 1) α ) α 1 ( 1) α (n 1) α 1 + λ = 0. We should oint out that Theorem 1.1 is known in the case = or L is the standard Lalacian. This is essentially roved by Yau [Y]. We refer to [L] for details. The second urose of the aer is to utilize the receding shar gradient estimate to study the manifolds whose rincial eigenvalue for the -Lalacian achieves its maximum value. Recall that the rincial eigenvalue λ 1 of the -Lalacian L is the maximum constant λ such that the equation L(v) = λ v 1 admits a ositive solution. Alternatively, λ 1 may be characterized variationally as the best constant of the following Poincaré inequality. λ 1 φ φ. A classical result of Cheng [C] says that λ 1 ( ) (n 1) on a comlete 4 n-dimensional manifold with Ric (n 1). The same argument also imlies λ 1 (L) ( ) n 1. This estimate is shar as demonstrated by Examle 1.3 and also the hyerbolic sace form H n. A natural question is to understand manifolds with λ 1 (L) achieving its maximum value. When =, this question has been studied by Li and the second author in [L-W1, L-W]. In articular, they have roved that such a manifold must be connected at infinity unless it is a toological cylinder 3
endowed with an exlicit wared roduct metric. We will take u the general case of 1 < < here and rovide a faithful generalization of their result. Our main conclusion can be summarized as follows. Theorem 1.4. Let n be a comlete manifold of dimension n 3 with Ric (n 1). If λ 1 (L) = ( ) n 1 for some (n 1), then either (n ) is connected at infinity or it slits as a wared roduct = R N with ds = dt + h (t)ds N, where N is comact, and the function h(t) = et if n 4 and h(t) = e t or h(t) = cosh t if n = 3. It is unclear to us at this oint whether the restriction on is necessary. It should also be emhasized that due to the nonlinear nature of the oerator L, a different aroach from [L-W1, L-W] is needed, and the shar gradient in Theorem 1.1 is crucial here. The following by-roduct of our argument seems worth mentioning. Corollary 1.5. Let (, g) be a comlete noncomact manifold with its Ricci curvature bounded below by a constant. If λ 1 (L) > 0 for some >, then λ 1 ( ) > 0 as well. In the last section, we rove a decay estimate in the sirit of Agmon [A] for the -Lalace equation. Agmon [A] has shown an eigenfunction corresonding to an eigenvalue below the essential sectrum of the Lalacian must decay exonentially rovided it does not grow too fast. An otimal version was later given in [L-W4]. Here, we consider the issue for the -Lalacian. However, our estimate is not as shar comared to the Lalacian case. Theorem 1.6. Let be a comlete Riemannian manifold and u a ositive function satisfying L(u) 0 on \ B(R 0 ), where B(R 0 ) is a geodesic ball in. Assume the rincial eigenvalue λ of L on \B(R 0 ) is ositive. If u satisfies the growth condition u ex( λ1/ 1 r) = o(r ), then for some constant C > 0. B(R) B(R+1)\B(R) u C ex( λ1/ 1 R) We would like to thank the referees for valuable suggestions to imrove the exosition of the aer. 4
Shar gradient estimate In this section, we rove a shar gradient estimate for ositive eigenfunctions of the -Lalacian. Recall the -Lalacian L is defined by L(v) = div( v v). A function v is an eigenfunction of -Lalacian with corresonding eigenvalue λ 0 if div( v v) = λ v v. We only consider ositive solutions v here and set u = ( 1) ln v. It can be easily verified that div( u u) = u + λ ( 1) 1. Denote L to be the linear oerator given by L(ϕ) = div ( f 1 A( ϕ) ), (.1) where f = u and A = id + ( ) f 1 u u. The following lemma is a direct calculation and due to [K-N]. Lemma.1. At oints where f > 0, L(f) = f 1 (u ij + R ij u i u j ) + ( 1) f f + f 1 u, f. Proof. Note that by the regularity theory v is smooth away from the oints where v = 0. By the definition of (.1), ) L(f) = f ( f 1 + ( ) div(f 1 u, f u + (f 1 ), f + ( ) f 1 u, f u = f 1 (u ij + R ij u i u j ) + ( 1)f f + ( )( )f 3 u, f + ( ) f (u ij f i u j + f ij u i u j ) + ( ) f u u, f + f 1 u, u. (.) 5
Since we have div ( u u ) = u + λ( 1) 1, u u + u, u = u + λ( 1) 1. (.3) Note that f = u. The equation (.3) becomes Therefore, f u + ( 1)f f, u = f + λ( 1) 1. (f ) u + ( 1)f f, u, u = f, u. This imlies ( )( )f 3 f, u + ( )f (f ij u i u j + u ij f i u j ) Combining with (.), we obtain Lemma.1. + ( )f u f, u + f 1 u, u = f 1 f, u. We are now ready to rove the gradient estimate. Theorem.. Let ( n, g) be an n-dimensional comlete noncomact manifold with Ric (n 1). Then for a ositive function v satisfying div( v v) = λv 1, ln v y, where y is the unique ositive root of the equation ( 1) y (n 1) y 1 + λ = 0. Proof. Let us first oint out that the value y > 0 is well defined since λ (n 1). As before, we let u = ( 1) ln v and f = u. The result in [W-Z] imlies that f c(n, ), a constant deending on and n. To obtain the shar estimate, let x be the unique ositive root of the equation x (n 1) x 1 + λ ( 1) 1 = 0. 6
For any δ > 0, consider ω = (f (x + δ)) +, that is, { f (x + δ), f > x + δ, ω = 0, Otherwise. Claim: L(ω) aω b w in the weak sense for some ositive constants a and b deending on, n and δ, that is, L(ϕ) ω ϕ (a ω b ω ) (.4) for any non-negative smooth function ϕ with comact suort on. Indeed, if we denote by = {f x + δ}, then L(ϕ) ω = = L(ϕ) ω ϕ L(ω) + f 1 A( ϕ), ν ω f 1 A( ω), ν ϕ, where ν is the outward unit normal vector of. Using ν = f = ω f and ω = 0 on, we conclude ω L(ϕ) ω = On the other hand, ϕ L(ω) + ϕ L(ω). f 1 ϕ A( ω), ω ω (.5) div ( u u ) = u + λ( 1) 1. (.6) Let {e 1, e,..., e n } be an orthonormal frame on with u e 1 = u. Then (.6) can be written into Therefore, n i,j=1 ( 1) u 11 + n u ii = f + u λ ( 1) 1. i= u ij ( n i= u ii) n + u 1i (.7) n 1 i=1 ( ) f + u λ ( 1) 1 ( 1) u 11 n + u n 1 1i. 7 i=1
Together with Lemma.1 and the Ricci curvature lower bound, on we have L(ω) { ( ) f 1 f + λ ( 1) 1 u ( 1) u 11 + n 1 + ( 1) f f + f 1 < u, ω > { ( f 1 f + λ ( 1) 1 u ) (n 1)f n 1 n + u ( 1) 1i n 1 f u λ ( } 1) 11 u u 11 n 1 i=1 +( 1) f f + f 1 < u, ω > n i=1 } u 1i (n 1)f f 1 ( )( ) f + λ ( 1) 1 u + (n 1) f 1 f + λ ( 1) 1 u (n 1) f 1 n 1 + 1 f f + f 1 < u, ω > ( 1) n 1 f 1 < u, ω > λ ( 1) n 1 where we have used the fact that u 11 = u, f f and n i=1 u 1i f 4 f. < u, ω >, f Since 0 < x f c(n, ) on and > 1, we conclude that, on L(ω) c 1 ( f (n 1)f 1 + λ( 1) 1) c ω, (.8) where c 1 and c are ositive constants deending on n and. We now verify that f (n 1)f 1 + λ( 1) 1 c 3 ω (.9) for some ositive constant c 3 deending on n, δ and. In fact, if we view both sides as a function of f, (.9) clearly holds true when f = x for any choice of c 3. Now the left hand side of (.9), as a function of 8
f, its derivative is given by f 1 (n 1)( 1) = 1 f 3 f 3 ( f 1 ( 1)(n 1) ). ( ) Note that λ ( n 1 ). So x ( 1) n 1. Using the fact that f x+δ, we conclude f 1 ( 1)(n 1) c(, n, δ) > 0. Therefore, (.9) holds true on. In articular, combining with (.8), we have L(ω) a ω b ω on, where a and b are ositive constants only deending on, n and δ. Plugging into (.5), we arrive at L(ϕ) ω ϕ L(ω) ϕ (a ω b ω ) = ϕ (a ω b ω ). In conclusion, L(ω) a ω b ω on in the weak sense as claimed. We now use the claim to show ω 0. Observe that for any cut-off function ϕ on and q 1, the function ϕ ω q may be used as a test function. So we have A( (ϕ ω q )), ω f 1 (a ϕ ω q+1 b ϕ ω q ω ). This imlies Note that a ϕ ω q+1 b ϕ ω q ω + c(n,, δ) ϕ ϕ ω ω q q ϕ ω q 1 ω, A( ω) f 1. 9
Thus, for any ɛ > 0, a ω, A( ω) = ω + ( ) u i u j u ω i ω j ϕ ω q+1 ɛ ϕ ω q+1 + b 4ɛ + c 4ɛ min{ 1, 1} ω. ϕ ω q 1 ω + ɛ ϕ ω q+1 ϕ ω q 1 ω c q ϕ ω q 1 ω, where c and c are constants deending on, n and δ. Choose q such that b + c = 4 ɛ c q. Then a ϕ ω q+1 ɛ ϕ ω q+1. Now let ϕ = 1 on B(k) and 0 outside B(k + 1) with ϕ, where k is a ositive integer. Then c ω q+1 ω q+1. ɛ B(k) B(k+1) Iterating this inequality, we conclude ( c k ω q+1 ɛ) B(k+1) B(1) This imlies either ω 0 or for all R 1, ω q+1 c 1 e R ln c ɛ B(R) ω q+1. for some ositive constants c 1 and c indeendent of ɛ. However, since ω is bounded and the volume of the ball B(R) satisfies V (R) c e (n 1)R, this leads to a contradiction if ɛ is chosen sufficiently small. Therefore, ω 0. In other words, f x as δ > 0 is arbitrary. This is obviously equivalent to ln v y. As ointed out reviously, this theorem is shar. We now draw a corollary that will be needed later. 10
Corollary.3. Let ( n, g) be an n-dimensional comlete noncomact manifold with Ric (n 1). Let v be a ositive solution of Then ln v n 1. div( v v) = ( n 1 ) v 1. 3 Structure at infinity In this section, we consider the manifolds with maximum value of λ 1 (L) and study its structure at infinity. We will first deal with the finite volume ends. Theorem 3.1. Let ( n, g) be a comlete manifold of dimension n. Suose that the Ricci curvature is bounded from below by Ric (n 1) and λ 1 (L) = ( n 1 ). Then either has no finite volume ends or n = R N n 1 with ds = dt + e t ds N and N being a comact manifold of nonnegative Ricci curvature. Proof. Suose has a finite volume end E. Let β be the Busemann function associated with a geodesic ray γ contained in E, that is, β(x) = lim t ( t d(x, γ(t) ). The Lalacian comarison theorem imlies Therefore, L ( e ) [ (n 1) β = div ( n 1 β (n 1). ) e (n 1) ( )β e n 1 β] = ( n 1 ) 1 e (n 1) ( 1)β β + ( n 1 ( n 1 ) 1 e [ (n 1) ( 1)β (n 1) 1 (n 1) ] = ( n 1 ) e (n 1) ( 1)β. So for v = e (n 1) β, we have L(v) λ 1 v 1. ) 1 ( n 1 )( 1)e n 1 ( 1)β β 11
Let ϕ be a nonnegative comactly suorted smooth function on. Then λ 1 (ϕ v) (ϕ v). Noting that ϕ v L(v) = and ϕ v ϕ 1 ϕ, v v v (ϕv) = ( ϕ v + ϕ v < ϕ, v > +ϕ v ) ϕ v + ϕ v < ϕ, v > ϕ v + c ϕ v for some constant c deending on, we conclude ϕ v ( L(v) + λ 1 v 1) (3.1) (ϕ v) ϕ v ϕ 1 ϕ, v v v (ϕ v) ϕ v ϕ 1 ϕ, v v v c ϕ v. = λ 1 Now choose ϕ = { 1, B(R), 0, \ B(R) with ϕ. Then R ϕ v = ϕ e (n 1)β (3.) 4 e (n 1)β R B(R)\B(R) = 4 e (n 1)β + 4 e (n 1)β. R R E (B(R)\B(R)) Since λ 1 (L) = ( n 1 ), by [B-K], we have V (E \ B(R)) c e (n 1)R. 1 (\E) (B(R)\B(R))
Thus, the first term goes to 0 as R goes to infinity. Note that (see [L-W3]) β(x) r(x) + c on \ E and V (B(R)) c e (n 1)R. The second term of (3.) converges to zero as well. Combining with (3.1), we conclude This imlies L(v) + λ 1 v 1 0. β = (n 1). The final conclusion that n = R N n 1 now follows from the argument in [L-W3]. We now turn to the infinite volume ends. Theorem 3.. Let ( n, g) be a comlete manifold of dimension n 3. Suose that the Ricci curvature is bounded from below by Ric (n 1) and λ 1 (L) = ( n 1 ) for some (n 1). Then has only one infinite (n ) volume end or n = R N n 1 for some comact manifold N with ds = dt + cosh (t) ds N. Proof. We may assume as otherwise by a simle Hölder inequality argument λ 1 ( ) = (n 1) and the theorem holds true by [L-W1]. 4 We first recall a general fact (see [L-W4]) that the existence of a ositive solution u to the equation u+ < h, u >= ρ u on imlies the validity of the following weighted Poincaré inequality ρ φ e h φ e h for any comactly suorted smooth function φ. Now let v be a ositive eigenfunction of the -Lalacian such that The equation can be re-written into div ( v v ) = λ 1 v 1. v + ( ) v, v v = λ v 1 v. v 13
Alying the aforementioned general fact with h = ( ) ln v and ρ = v λ 1, we conclude v v λ 1 v ϕ v ϕ v for any comactly suorted smooth function ϕ on. Obviously, the inequality can be simlified into λ 1 ϕ v ϕ v. Now let w = v. Then ( ϕ ) λ 1 ϕ = λ 1 v w Exanding the right hand side of (3.3), we get = = Note that ( ϕ ) v w ϕ w ϕ w w { ϕ v v v ( ϕ ) v. (3.3) w + ϕ w v ϕ ϕ, w v }. w 4 w 3 Hence w v = ( w 4 ) v v. λ 1 ϕ ( ϕ v + ( ) v ) ϕ v v + 1 ϕ, w v. (3.4) The last term of (3.4) can be simlified as follows. 14
Using or 1 ϕ, w v = 1 ϕ (w ) v 1 ϕ (w ), v = 1 ) ϕ ( v (w ) + (w ), ( v ). (3.5) div ( v v ) = λ 1 v 1 we obtain v, v + v v = λ 1 v 1, v (v ) + (v ), ( v ) = v ( ) v 1 v + v ( )(1 ) v v +( ) v 1 v, v = ( ) v 1 ( λ 1 v 1 ) + ( )(1 ) v v = λ 1 ( ) + ( )( 1) v v. (3.6) Plugging (3.6) into (3.5), we conclude 1 ϕ, w v = 1 ϕ ( λ 1 ( ) + ( )( 1) v v ). (3.7) Putting (3.7) into (3.4) and collecting terms, we arrive at ( λ 1 ϕ ( ) ) + ϕ v ϕ v. (3.8) 4 v v By Corollary.3, v v n 1 (n 1) as λ 1 = ( n 1 ). Therefore, ϕ ϕ. 15
This shows that λ 1 ( ) (n 1). Now by the result in [L-W1], if (n 1) n or (n 1), then either has only one infinite volume end or (n ) n = R N n 1 with N being comact and ds = dt + cosh (t) ds N. From (3.8) and Theorem., one can easily conclude the following. Corollary 3.3. Let (, g) be a comlete noncomact manifold with its Ricci curvature bounded below by a constant. If λ 1 (L) > 0 for some >, then λ 1 ( ) > 0 as well. 4 Decay estimate In this section, we rove a decay estimate for subsolutions to the -Lalace equation in the sirit of Agmon [A]. Theorem 4.1. Let be a comlete Riemannian manifold and u a ositive function satisfying L(u) µ u 1 on \ B(R 0 ), where B(R 0 ) is a geodesic ball in and µ a constant strictly smaller than the rincial eigenvalue λ of L on \ B(R 0 ). If u satisfies the growth condition u ex( λ µ (r)) = o(r ), B(R) 1λ 1 then B(R+1)\B(R) for some constant C > 0. u C ex( λ µ 1λ 1 (R)) Proof. For a comactly suorted smooth function ϕ on \ B(R 0 ), the variational characterization for λ imlies λ (ϕ u) ( 3 ɛ (ϕu) ) 1 for any 0 < ɛ < 1. On the other hand, we have ϕ u + (1 + ɛ) ϕ u (4.1) 16
Thus, µ ϕ u ϕ u div( u u) = (ϕ u), u u ) = (ϕ u + ϕ 1 ϕ, u u u. ϕ u 1 ɛ 1 ϕ 1 ϕ u 1 u + µ ϕ u ϕ u + ɛ ϕ u + µ ϕ u. So we get (1 ɛ) ϕ u 1 ϕ u + µ ϕ u. (4.) ɛ 1 Putting (4.) into (4.1) and otimizing over ɛ, we conclude (λ µ) (ϕ u) 6 (1λ ) 1 ϕ u. (4.3) Let us now choose ϕ = φ e h, where and r(x) R 0, B(R 0 + 1) \ B(R 0 ), 1, B(R) \ B(R 0 + 1), φ = R r(x), B(R) \ B(R), R 0, \ B(R) h = { δ r(x), K δ r(x), r K δ r K δ for some fixed K, where δ = λ µ. Substituting into (4.3), we obtain 1 λ 1 (λ µ) (φe h u) 6 (1λ ) 1 (φ e h ) u 6 (1λ ) 1 17 ( (3) 1 φ e h u + φ u δ e h).
Noting that 1 (1λ ) 1 δ = (λ µ) < (λ µ), we have (λ µ) (1 1 ) c() B(R 0 +1)\B(R 0 ) B( K δ )\B(R 0) e δr u + φ e δ r u c(, K) R B(R)\B(R) e δr u. Using the growth assumtion on u, one sees the last term goes to 0 as R. Therefore, by first letting R and then K, we arrive at e δr u <. The theorem is roved. \B(R 0 ) References [A] S. Agmon, Lectures on exonential decay of solutions of second-order ellitic equations: bounds on eigenfunctions of N-body Schrödinger oerators, athematical Notes, 9, Princeton University Press, 198 [B-K] S. Buckley and P. Koskela, Ends of metric measure saces and Sobolev inequality, ath. Z. 5 (005), 75 85 [C] [H-I] S.Y. Cheng, Eigenvalue comarison theorems and its geometric alications, ath. Z. 143 (1975), 89 97. G. Huisken and T. Ilmanen, The inverse mean curvature flow and Riemannian Penrose inequality, J. Differential Geom. 59 (001), 353 437. [K-N] B. Kotschwar and L. Ni, Local gradient estimates of -harmonic functions, -flow, and an entroy formula, Ann Sci École Norm. Su., 4 1 H (009), 1 36 [L] P. Li, Geometric Analysis, Cambridge Studies in Advanced athematics, 01. [L-W1] P. Li and J. Wang, Comlete manifolds with ositive sectrum, J. Differential Geom. 58 (001) 501-534. [L-W] P. Li and J. Wang, Comlete manifolds with ositive sectrum, II, J. Differential Geom. 6 (00), 143 16. 18
[L-W3] P. Li and J. Wang, Connectedness at infinity of comlete Kähler manifolds, Amer. J. ath. 131 (009), 771 817. [L-W4] P. Li and J. Wang, Weighted Poincaré inequality and rigidity of comlete manifolds, Ann. Sci. École. Norm. Su. (4) 39 (006), 91 98. [] R. oser, The inverse mean curvature flow and -harmonic functions, J. Euro. ath. Soc., 9 (007), 77 83. [W-Z] X. Wang and L. Zhang, Local gradient estimate for -harmonic functions on Riemannian manifolds, Comm. Anal. Geom. 19 (011), 759 77. [Y] S.T. Yau, Harmonic functions on comlete Riemannian manifolds, Comm. Pure Al. ath. 8 (1975), 01 8. Deartment of athematics, National Tsing Hua University, Hsin-Chu, Taiwan 30013 E-mail address: cjsung@math.nthu.edu.tw School of athematics, University of innesota, inneaolis, N 55455 E-mail address: jiaing@math.umn.edu 19