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Gradient estimates for positive smooth f-harmonic functions
Acta Mathematica Scientia 2010,30B(5):1614–1618 http://actams.wipm.ac.cn
GRADIENT ESTIMATES FOR POSITIVE SMOOTH f -HARMONIC FUNCTIONS∗ Chen Li (
)
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China E-mail: chernli163.com
Chen Wenyi (
)
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China E-mail: [email protected]
Abstract For Riemannian manifolds with a measure, we study the gradient estimates for positive smooth f -harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below, generalizing the classical ones of Yau (i.e., when f is constant). Key words gradient estimate; f -harmonic function; Bakry-Emery Ricci tensor 2000 MR Subject Classification
1
35K05; 58J35
Introduction and Main Results
In this article, we study smooth metric measure spaces (M n , g, e−f dvolg ), where g is a smooth complete metric on an n-dimensional manifold M , f is a smooth real valued function on M and dvolg is the Riemannian volume density on M . The Bakry-Emery Ricci tensor is Ricf = Ric + D2 f, where D2 f is the Hessian matrix of the function f and Ric is the Ricci tensor of the metric g. This is often also referred to the ∞-Bakry-Emery Ricci tensor. Bakry and Emery [2] extensively studies this tensor and its relationship to diffusion processes. The Bakry-Emery tensor also occurs naturally in many different subjects [5, 8]. The equation Ricf = λg for some constant λ is exactly the gradient Ricci soliton equation, which plays an important role in the theory of Ricci flow. Moreover Ricf has a natural extension to metric measure spaces [6, 9, 10]. The f -Laplacian of a function u is defined by Δf u = Δu − D∇f u = Δu − ∇f, ∇u, where Δ is the standard Laplace-Beltrami Operator. A function u is called f -harmonic if Δf u = 0. In smooth metric measure spaces (M, g, e−f dvolg ), the operator Δf is self-adjoint. ∗ Received
May 6, 2008. The research is supported by NSFC (10471108, 10631020).
L. Chen & W.Y. Chen: GRADIENT ESTIMATES FOR f -HARMONIC FUNCTIONS
No.5
Namely, the following identity holds for Δf (φ)ψdvolg = M
M
1615
φΔf ψdvolg .
The measure dvolg also plays an important role in Perelman’s entropy formulas for the Ricci flow [7]. When f is a constant function, the Bakry-Emery Ricci tensor is the Ricci tensor, and the f -Laplacian is the standard Laplace-Beltrami Operator. For positive smooth harmonic functions on complete Riemannian manifolds, there is well-known gradient estimates of Yau [12]. Therefore, it is natural to investigate whether this results can extend to positive smooth f -harmonic functions for the Bakry-Emery Ricci tensor. Theorem Let (M n , g, e−f dvolg ) be a smooth metric measure space with Ricf ≥ −H (H ≥ 0). Assume Ric ≥ −K(K ≥ 0). If u is a positive f -harmonic function on the geodesic ball Bp (2R). Then, in the geodesic ball Bp (R), √ |∇u| 1 ≤ C(n) H + K + sup |∇f | + , u R B2R (p) where the constant C(n) depends only on n. Remark If f = const., our results are reduced to Yau’s results of gradient estimates for positive harmonic functions on complete Riemannian manifolds. Therefore, we generalize Yau’s results. Corollary (Harnack inequality) Let (M n , g, e−f dvolg ) be a smooth metric measure space with Ricf ≥ −H(H ≥ 0) . Assume Ric≥ −K(K ≥ 0). If u is a positive f -harmonic function on the geodesic ball Bp (2R), then, √ 1 sup u, sup u ≤ C(n) H + K + sup |∇f | + R B R (p) B R (p) B2R (p) 2
2
where the constant C(n) depends only on n.
2
Proof of Theorems First, we recall the following Weitzenbock-Bochner formula [11, 13] for Δf : Lemma 2.1 For a smooth function w on M , it holds that 1 Δf |∇w|2 = |D2 w|2 + ∇w, ∇Δf w + Ricf (∇w, ∇w). 2 Proof Using the Bochner formula for w, we get 1 Δ|∇w|2 = |D2 w|2 + ∇w, ∇Δw + Ric(∇w, ∇w). 2
Since
and
1 1 1 Δf |∇w|2 = Δ|∇w|2 − ∇f, ∇|∇w|2 2 2 2 1 ∇w, ∇∇f, ∇w = D2 f (∇w, ∇w) + ∇f, ∇|∇w|2 , 2
1616
ACTA MATHEMATICA SCIENTIA
Vol.30 Ser.B
we get the result. Lemma 2.2 Let (M n , g, e−f dvolg ) be a smooth metric measure space with Ricf ≥ −H (H ≥ 0), u is a positive smooth f -harmonic function on M . Then, we have the estimate: 3 1 1 2 1 ΔF ≥ F 2 − HF − |∇f |F 2 − ∇w, ∇F + ∇f, ∇F , (2.1) 2 n n 2 where we denote F = |∇ log u|2 . Proof Set w = log u, then w satisfies the equation Δf w + |∇w|2 = 0.
(2.2)
Using the Weitzenbock-Bochner formula for Δf , we get 1 Δf |∇w|2 = |D2 w|2 + ∇w, ∇Δf w + Ricf (∇w, ∇w). 2 Let F = |∇w|2 , then using equation (2.2), we have Δf w = −F. Therefore, |D2 w|2 ≥
1 1 1 2 |Δw|2 = |F − ∇f, ∇w|2 ≥ F 2 − ∇f, ∇wF n n n n
and ∇w, ∇Δf w = −∇w, ∇F . Hence, 1 1 2 Δf F ≥ F 2 − ∇f, ∇wF − ∇w, ∇F + Ricf (∇w, ∇w). 2 n n Using Ricf ≥ −H, we get 1 1 1 2 ΔF ≥ F 2 − HF − ∇f, ∇wF − ∇w, ∇F + ∇f, ∇F . 2 n n 2 Rewriting this and using the Cauchy-Schwarz inequality, we get 1
∇f, ∇w ≤ |∇f ||∇w| = |∇f |F 2 . We find that 1 3 1 2 1 ΔF ≥ F 2 − HF − |∇f |F 2 − ∇w, ∇F + ∇f, ∇F . 2 n n 2 Thus the estimate follows. Now, we prove the main theorem. Proof Let ψ be a C ∞ function defined on [0, ∞), such that ψ(r) = 1 for r ≤ 1, ψ(r) = 0 for r ≥ 2, 0 ≤ ψ(r) ≤ 1, and 1
0 ≥ ψ − 2 ψ ≥ −c1 , ψ ≥ −c2 ,
No.5
L. Chen & W.Y. Chen: GRADIENT ESTIMATES FOR f -HARMONIC FUNCTIONS
1617
where c1 and c2 are certain positive constants. If d(p, x) denotes the geodesic distance between p and x, we set d(p, x) . φ(x) = ψ R Using an argument of Calabi [3] (see also Cheng and Yau [4]), we can assume, without loss of generality, that φ(x) is smooth in Bp (2R). Then, we have the inequalities |∇φ|2 c2 ≤ 12 φ R and
√ (n − 1)c21 (1 + R K) + c2 , Δφ ≥ − R2 where the last inequality is obtained by the Laplacian comparison theorem (see [1]). Let x0 be the point in Bp (2R), at which φF takes its maximum value Λ. Then at the point x0 , ∇(φF ) = 0, Δ(φF ) ≤ 0. Therefore, at x0 φΔF + F Δφ − 2F |∇φ|2 φ−1 ≤ 0. So for
√ 2c21 + (n − 1)c21 (1 + R K) + c2 , A= R2
we have φΔF ≤ AF.
(2.3)
Since at x0 ,
1
−φ∇f, ∇F = ∇f, ∇φF ≤ F |∇f |φ 2
c1 R
and
c1 . R Then using (2.1) and (2.3) at x0 and multiplying both sides by φ, we have 3
1
φ∇w, ∇F = −∇w, ∇φF ≤ F 2 φ 2
1 3 3 3 c1 1 1 2 c1 1 AΛ ≥ Λ2 − HφΛ − |∇f |φ 2 Λ 2 − φΛ 2 − |∇f |φ 2 Λ. 2 n n R 2 R
That is,
1
2 3 c1 1 1 1 c1 3 A + Hφ + |∇f |φ 2 Λ ≥ Λ2 − |∇f |φ 2 + φ Λ 2 . 2 2 R n n R Using the Cauchy-Schwarz inequality, we obtain 2 n Therefore,
Then,
1
|∇f |φ 2 +
c1 3 1 2 2 nc1 2 1 |∇f |φ 2 + φ Λ2 ≤ Λ + φ Λ. R 2n n 2R
1
3 c1 1 1 2 1 2 nc1 2 A + Hφ + |∇f |φ 2 + |∇f |φ 2 + φ Λ . Λ≥ 2 2 R n 2R 2n
1 Λ ≤ C(n) H + K + |∇f |2 + 2 . R
1618 Hence,
ACTA MATHEMATICA SCIENTIA
Vol.30 Ser.B
1 sup F ≤ C(n) H + K + sup |∇f |2 + 2 . R BR (p) B2R (p)
Therefore, we confirm our result. Finally, we prove the corollary stated at the end of Section 1. Proof The proof is the same as Yau’s Harnack inequality for positive harmonic functions on complete Riemannian manifolds [11]. References [1] Aubin T. Non-linear Analysis on Manifolds. New York: Springer, 1982 ´ [2] Bakry D, Emery M. Diffusions hypercontractives. S´eminaire de Probabilit´es, XIX, 1983/1984, 1985, 1123: 177–206 [3] Calabi E. An extension of E. Hopf’s maximum principle with application to Riemannian geometry. Duke J Math, 1957, 25:45–46 [4] Cheng S Y, Yau S T. Differential equations on Riemannian manifolds and their geomtric applications. Comm Pure Apple Math, 1975, 28: 333–354 [5] Lott J. Some geometric properties of the Bakry-Emery-Ricci tensor. Comment Math Helv, 2003, 78(4): 865–833 [6] Lott J, Villani C. Ricci curvature for metric-measure spaces via optimal transport. Ann Math, 2009, 169(3): 903–991 [7] Perelman G Y. Ricci flow with surgery on three manifilds. arXiv:math/0303109, 2003 [8] Petersen P, Wylie W. Rigidity of gradient Ricci soliton. Pacific J Math, 2009, 24(1): 329–345 [9] Sturm K T. On the geometry of metric measure spaces I. Acta Math, 2006, 196(1): 65–131 [10] Sturm K T. On the geometry of metric measure spaces II. Acta Math, 2006, 196(1): 133–177 [11] Wei G F, Wylie W. Comparison geometry for the Bakry-Emery Ricci tensor. J Diff Geom, 2009, 83(2): 337–405 [12] Yau S T, Schoen R. Lectures on Differential Geometry. Boston: International Press, 1994 [13] Zhong T D, Zhong C P. Bochner technique in real Finsler Manifolds. Acta Math Sci, 2003, 23(2): 165–177
GRADIENT ESTIMATES FOR POSITIVE SMOOTH f -HARMONIC FUNCTIONS∗ Chen Li (
)
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China E-mail: chernli163.com
Chen Wenyi (
)
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China E-mail: [email protected]
Abstract For Riemannian manifolds with a measure, we study the gradient estimates for positive smooth f -harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below, generalizing the classical ones of Yau (i.e., when f is constant). Key words gradient estimate; f -harmonic function; Bakry-Emery Ricci tensor 2000 MR Subject Classification
1
35K05; 58J35
Introduction and Main Results
In this article, we study smooth metric measure spaces (M n , g, e−f dvolg ), where g is a smooth complete metric on an n-dimensional manifold M , f is a smooth real valued function on M and dvolg is the Riemannian volume density on M . The Bakry-Emery Ricci tensor is Ricf = Ric + D2 f, where D2 f is the Hessian matrix of the function f and Ric is the Ricci tensor of the metric g. This is often also referred to the ∞-Bakry-Emery Ricci tensor. Bakry and Emery [2] extensively studies this tensor and its relationship to diffusion processes. The Bakry-Emery tensor also occurs naturally in many different subjects [5, 8]. The equation Ricf = λg for some constant λ is exactly the gradient Ricci soliton equation, which plays an important role in the theory of Ricci flow. Moreover Ricf has a natural extension to metric measure spaces [6, 9, 10]. The f -Laplacian of a function u is defined by Δf u = Δu − D∇f u = Δu − ∇f, ∇u, where Δ is the standard Laplace-Beltrami Operator. A function u is called f -harmonic if Δf u = 0. In smooth metric measure spaces (M, g, e−f dvolg ), the operator Δf is self-adjoint. ∗ Received
May 6, 2008. The research is supported by NSFC (10471108, 10631020).
L. Chen & W.Y. Chen: GRADIENT ESTIMATES FOR f -HARMONIC FUNCTIONS
No.5
Namely, the following identity holds for Δf (φ)ψdvolg = M
M
1615
φΔf ψdvolg .
The measure dvolg also plays an important role in Perelman’s entropy formulas for the Ricci flow [7]. When f is a constant function, the Bakry-Emery Ricci tensor is the Ricci tensor, and the f -Laplacian is the standard Laplace-Beltrami Operator. For positive smooth harmonic functions on complete Riemannian manifolds, there is well-known gradient estimates of Yau [12]. Therefore, it is natural to investigate whether this results can extend to positive smooth f -harmonic functions for the Bakry-Emery Ricci tensor. Theorem Let (M n , g, e−f dvolg ) be a smooth metric measure space with Ricf ≥ −H (H ≥ 0). Assume Ric ≥ −K(K ≥ 0). If u is a positive f -harmonic function on the geodesic ball Bp (2R). Then, in the geodesic ball Bp (R), √ |∇u| 1 ≤ C(n) H + K + sup |∇f | + , u R B2R (p) where the constant C(n) depends only on n. Remark If f = const., our results are reduced to Yau’s results of gradient estimates for positive harmonic functions on complete Riemannian manifolds. Therefore, we generalize Yau’s results. Corollary (Harnack inequality) Let (M n , g, e−f dvolg ) be a smooth metric measure space with Ricf ≥ −H(H ≥ 0) . Assume Ric≥ −K(K ≥ 0). If u is a positive f -harmonic function on the geodesic ball Bp (2R), then, √ 1 sup u, sup u ≤ C(n) H + K + sup |∇f | + R B R (p) B R (p) B2R (p) 2
2
where the constant C(n) depends only on n.
2
Proof of Theorems First, we recall the following Weitzenbock-Bochner formula [11, 13] for Δf : Lemma 2.1 For a smooth function w on M , it holds that 1 Δf |∇w|2 = |D2 w|2 + ∇w, ∇Δf w + Ricf (∇w, ∇w). 2 Proof Using the Bochner formula for w, we get 1 Δ|∇w|2 = |D2 w|2 + ∇w, ∇Δw + Ric(∇w, ∇w). 2
Since
and
1 1 1 Δf |∇w|2 = Δ|∇w|2 − ∇f, ∇|∇w|2 2 2 2 1 ∇w, ∇∇f, ∇w = D2 f (∇w, ∇w) + ∇f, ∇|∇w|2 , 2
1616
ACTA MATHEMATICA SCIENTIA
Vol.30 Ser.B
we get the result. Lemma 2.2 Let (M n , g, e−f dvolg ) be a smooth metric measure space with Ricf ≥ −H (H ≥ 0), u is a positive smooth f -harmonic function on M . Then, we have the estimate: 3 1 1 2 1 ΔF ≥ F 2 − HF − |∇f |F 2 − ∇w, ∇F + ∇f, ∇F , (2.1) 2 n n 2 where we denote F = |∇ log u|2 . Proof Set w = log u, then w satisfies the equation Δf w + |∇w|2 = 0.
(2.2)
Using the Weitzenbock-Bochner formula for Δf , we get 1 Δf |∇w|2 = |D2 w|2 + ∇w, ∇Δf w + Ricf (∇w, ∇w). 2 Let F = |∇w|2 , then using equation (2.2), we have Δf w = −F. Therefore, |D2 w|2 ≥
1 1 1 2 |Δw|2 = |F − ∇f, ∇w|2 ≥ F 2 − ∇f, ∇wF n n n n
and ∇w, ∇Δf w = −∇w, ∇F . Hence, 1 1 2 Δf F ≥ F 2 − ∇f, ∇wF − ∇w, ∇F + Ricf (∇w, ∇w). 2 n n Using Ricf ≥ −H, we get 1 1 1 2 ΔF ≥ F 2 − HF − ∇f, ∇wF − ∇w, ∇F + ∇f, ∇F . 2 n n 2 Rewriting this and using the Cauchy-Schwarz inequality, we get 1
∇f, ∇w ≤ |∇f ||∇w| = |∇f |F 2 . We find that 1 3 1 2 1 ΔF ≥ F 2 − HF − |∇f |F 2 − ∇w, ∇F + ∇f, ∇F . 2 n n 2 Thus the estimate follows. Now, we prove the main theorem. Proof Let ψ be a C ∞ function defined on [0, ∞), such that ψ(r) = 1 for r ≤ 1, ψ(r) = 0 for r ≥ 2, 0 ≤ ψ(r) ≤ 1, and 1
0 ≥ ψ − 2 ψ ≥ −c1 , ψ ≥ −c2 ,
No.5
L. Chen & W.Y. Chen: GRADIENT ESTIMATES FOR f -HARMONIC FUNCTIONS
1617
where c1 and c2 are certain positive constants. If d(p, x) denotes the geodesic distance between p and x, we set d(p, x) . φ(x) = ψ R Using an argument of Calabi [3] (see also Cheng and Yau [4]), we can assume, without loss of generality, that φ(x) is smooth in Bp (2R). Then, we have the inequalities |∇φ|2 c2 ≤ 12 φ R and
√ (n − 1)c21 (1 + R K) + c2 , Δφ ≥ − R2 where the last inequality is obtained by the Laplacian comparison theorem (see [1]). Let x0 be the point in Bp (2R), at which φF takes its maximum value Λ. Then at the point x0 , ∇(φF ) = 0, Δ(φF ) ≤ 0. Therefore, at x0 φΔF + F Δφ − 2F |∇φ|2 φ−1 ≤ 0. So for
√ 2c21 + (n − 1)c21 (1 + R K) + c2 , A= R2
we have φΔF ≤ AF.
(2.3)
Since at x0 ,
1
−φ∇f, ∇F = ∇f, ∇φF ≤ F |∇f |φ 2
c1 R
and
c1 . R Then using (2.1) and (2.3) at x0 and multiplying both sides by φ, we have 3
1
φ∇w, ∇F = −∇w, ∇φF ≤ F 2 φ 2
1 3 3 3 c1 1 1 2 c1 1 AΛ ≥ Λ2 − HφΛ − |∇f |φ 2 Λ 2 − φΛ 2 − |∇f |φ 2 Λ. 2 n n R 2 R
That is,
1
2 3 c1 1 1 1 c1 3 A + Hφ + |∇f |φ 2 Λ ≥ Λ2 − |∇f |φ 2 + φ Λ 2 . 2 2 R n n R Using the Cauchy-Schwarz inequality, we obtain 2 n Therefore,
Then,
1
|∇f |φ 2 +
c1 3 1 2 2 nc1 2 1 |∇f |φ 2 + φ Λ2 ≤ Λ + φ Λ. R 2n n 2R
1
3 c1 1 1 2 1 2 nc1 2 A + Hφ + |∇f |φ 2 + |∇f |φ 2 + φ Λ . Λ≥ 2 2 R n 2R 2n
1 Λ ≤ C(n) H + K + |∇f |2 + 2 . R
1618 Hence,
ACTA MATHEMATICA SCIENTIA
Vol.30 Ser.B
1 sup F ≤ C(n) H + K + sup |∇f |2 + 2 . R BR (p) B2R (p)
Therefore, we confirm our result. Finally, we prove the corollary stated at the end of Section 1. Proof The proof is the same as Yau’s Harnack inequality for positive harmonic functions on complete Riemannian manifolds [11]. References [1] Aubin T. Non-linear Analysis on Manifolds. New York: Springer, 1982 ´ [2] Bakry D, Emery M. Diffusions hypercontractives. S´eminaire de Probabilit´es, XIX, 1983/1984, 1985, 1123: 177–206 [3] Calabi E. An extension of E. Hopf’s maximum principle with application to Riemannian geometry. Duke J Math, 1957, 25:45–46 [4] Cheng S Y, Yau S T. Differential equations on Riemannian manifolds and their geomtric applications. Comm Pure Apple Math, 1975, 28: 333–354 [5] Lott J. Some geometric properties of the Bakry-Emery-Ricci tensor. Comment Math Helv, 2003, 78(4): 865–833 [6] Lott J, Villani C. Ricci curvature for metric-measure spaces via optimal transport. Ann Math, 2009, 169(3): 903–991 [7] Perelman G Y. Ricci flow with surgery on three manifilds. arXiv:math/0303109, 2003 [8] Petersen P, Wylie W. Rigidity of gradient Ricci soliton. Pacific J Math, 2009, 24(1): 329–345 [9] Sturm K T. On the geometry of metric measure spaces I. Acta Math, 2006, 196(1): 65–131 [10] Sturm K T. On the geometry of metric measure spaces II. Acta Math, 2006, 196(1): 133–177 [11] Wei G F, Wylie W. Comparison geometry for the Bakry-Emery Ricci tensor. J Diff Geom, 2009, 83(2): 337–405 [12] Yau S T, Schoen R. Lectures on Differential Geometry. Boston: International Press, 1994 [13] Zhong T D, Zhong C P. Bochner technique in real Finsler Manifolds. Acta Math Sci, 2003, 23(2): 165–177