Semigroup Properties

for the Second Fundamental Form

Feng-Yu Wang^{1}

Received: April 12, 2009

Communicated by Friedrich Götze

Abstract. Let M be a compact Riemannian manifold with boundary ∂M and L = δ+Z for a C^{1}-vector field Z on M. Several equivalent statements, including the gradient and Poincaré/log-Sobolev type inequalities of the Neumann semigroup generated by L, are presented for lower bound conditions on the curvature of L and the second fundamental form of ∂M. The main result not only generalizes the corresponding known ones on manifolds without boundary, but also clarifies the role of the second fundamental form in the analysis of the Neumann semigroup. Moreover, the Lévy-Gromov isoperimetric inequality is also studied on manifolds with boundary.

2010 Mathematics Subject Classification: 60J60, 58G32. Keywords and Phrases: Second fundamental form, gradient estimate, Neumann semigroup, log-Sobolev inequality, Poincaré inequality.

1 Introduction

The main purpose of this paper is to find out equivalent properties of the Neumann semigroup on manifolds with boundary for lower bounds of the second fundamental form of the boundary. To explain the main idea of the study, let us briefly recall some equivalent semigroup properties for curvature lower bounds on manifolds without boundary. Let M be a connected complete Riemannian manifold without boundary and let L = Δ + Z for some C^{1}-vector field Z on M. Let P_{t} be the diffusion semigroup generated by L, which is unique and Markovian if the curvature of L is bounded below, namely (see [3]),

Ric − ∇Z ≥ −K  (1.1)

1Supported in part by NNSFC(10721091) and the 973-Project.

Documenta Mathematica 15 (2010) 527–543

holds on M for some constant K ∈ ℝ. The following is a collection of known equivalent statements for (1.1), where the first two ones on gradient estimates are classical in geometry (see e.g. [1, 5, 6, 7]), and the remainder follows from Propositions 2.1 and 2.6 in [2] (see also [9]):

(i) |∇P_{t}f|^{2} ≤ e^{2Kt}P_{t}|∇f|^{2}, t ≥ 0, f ∈ C_{b}^{1}(M);

(ii) |∇P_{t}f| ≤ e^{Kt}P_{t}|∇f|, t ≥ 0, f ∈ C_{b}^{1}(M);

(iii) P_{t}f^{2} − (P_{t}f)^{2} ≤ \frac{e^{2Kt} − 1}{K}P_{t}|∇f|^{2}, t ≥ 0, f ∈ C_{b}^{1}(M);

(iv) P_{t}f^{2} − (P_{t}f)^{2} ≥ \frac{1 − e^{−2Kt}}{K} |∇P_{t}f|^{2}, t ≥ 0, f ∈ C_{b}^{1}(M);

(v) P_{t}(f^{2} log f^{2}) − (P_{t}f^{2}) log(P_{t}f^{2}) ≤ \frac{2(e^{2Kt} − 1)}{K}P_{t}|∇f|^{2}, t ≥ 0, f ∈

C_{b}^{1}(M);

(vi) (P_{t}f)\{P_{t}(f log f) − (P_{t}f) log(P_{t}f)\} ≥ \frac{1 − e^{−2Kt}}{2K} |∇P_{t}f|^{2}, t ≥ 0, f ∈

C_{b}^{1}(M), f ≥ 0.

These equivalent statements for the curvature condition are crucial in the study of heat semigroups and functional inequalities on manifolds. For the case that M has a convex boundary, these equivalences are also true for P_{t} the Neumann semigroup (see [10] for one more equivalent statement on Harnack inequality). The question is now can we extend this result to manifolds with non-convex boundary, and furthermore describe the second fundamental using semigroup properties? So, from now on we assume that M has a boundary ∂M. Let N be the inward unit normal vector field on ∂M. Then the second fundamental form is a twotensor on T ∂M, the tangent space of ∂M, defined by

𝕀(X, Y) = −⟨∇_{X}N, Y ⟩, X, Y ∈ T ∂M.

If 𝕀 ≥ 0(i.e. 𝕀(X, X) ≥ 0 for X ∈ T ∂M), then ∂M (or M) is called convex. In general, we intend to study the lower bound condition of 𝕀; namely, 𝕀 ≥ −σ on ∂M for some σ ∈ ℝ. For x ∈ M, let 𝔼^{x} be the expectation taken for the reflecting L-diffusion process X_{t} starting from x. So, for a bounded measurable functional Φ of X,

𝔼Φ : x ↦ 𝔼^{x}Φ

is a function on M. Moreover, let l_{t} be the local time of X_{t} on ∂M. According to [8, Theorem 5.1], (1.1) and 𝕀 ≥ −σ imply

|∇P_{t}f| ≤ e^{Kt}𝔼[|∇f|(X_{t})|e^{σl_{t}}], t > 0, f ∈ C^{1}(M).  (1.2)

Semigroup Properties for the Second Fundamental Form 529

To see that (1.2) is indeed equivalent to (1.1) and 𝕀 ≥ −σ, we shall make use of the following formula for the second fundamental form established recently by the author in [12]: for any f ∈ C^{∞}(M) satisfying the Neumann condition Nf|_{∂M} = 0,

𝕀(∇f, ∇f) = \frac{\sqrt{π}|∇f|^{2}}{2} lim_{t→0} \frac{1}{\sqrt{t}} log \frac{(P_{t}|∇f|^{p})^{1/p}}{|∇P_{t}f|}  (1.3)

holds on ∂M for any p ∈ [1, ∞). With help of this result and stochastic analysis on the reflecting diffusion process, we are able to prove the following main result of the paper.

Theorem 1.1. Let M be a compact Riemannian manifold with boundary and let P_{t} be the Neumann semigroup generated by L = Δ + Z. Then for any constants K, σ ∈ ℝ, the following statements are equivalent to each other:

(1) Ric − ∇Z ≥ −K on M and 𝕀 ≥ −σ on ∂M;

(2) (1.2) holds;

(3) |∇P_{t}f|^{2} ≤ e^{2Kt}(P_{t}|∇f|^{2})𝔼e^{2σl_{t}}, t ≥ 0, f ∈ C^{1}(M);

(4) P_{t}(f^{2} log f^{2}) − (P_{t}f^{2}) log P_{t}f^{2} ≤ 4𝔼[|∇f|^{2}(X_{t}) ∫_{0}^{t} e^{2σ(l_{t}−l_{t−s})+2Ks}ds], t ≥ 0, f ∈ C^{1}(M);

(5) P_{t}f^{2} −(P_{t}f)^{2} ≤ 2𝔼[|∇f|^{2}(X_{t}) ∫_{0}^{t} e^{2σ(l_{t}−l_{t−s})+2Ks}ds], t ≥ 0, f ∈ C^{1}(M);

(6) |∇P_{t}f|^{2} ≤

≤ (\frac{2K}{1 − e^{−2Kt}} )^{2}(P_{t}(f log f) − (P_{t}f) log P_{t}f)𝔼[f(X_{t}) ∫_{0}^{t} e^{2σl_{s}−2Ks}ds],

t > 0, f ≥ 0, f ∈ C^{1}(M);

(7) |∇P_{t}f|^{2} ≤ \frac{2K^{2}}{(1 − e^{−2Kt})^{2}} (P_{t}f^{2} − (P_{t}f)^{2})𝔼 ∫_{0}^{t} e^{2σl_{s}−2Ks}ds, t ≥ 0, f ∈

C^{1}(M).

Theorem 1.1 can be extended to a class of non-compact manifolds with boundary such that the local times l_{t} is exponentially integrable. According to [13] the later is true provided 𝕀 is bounded, the sectional curvature around ∂M is bounded above, the drift Z is bounded around ∂M, and the injectivity radius of the boundary is positive. To avoid technical complications, here we simply consider the compact case. In the next section, we shall provide a result on gradient estimate and nonconstant lower bounds of curvature and second fundamental form, which implies the equivalences among (1), (2) and (3) as a special case. Then we present a complete proof for the remainder of Theorem 1.1 in Section 3. As mentioned above, for manifolds without boundary or with a convex boundary an equivalent Harnack inequality for the curvature condition has been presented in [10].

Due to unboundedness of the local time which causes an essential difficulty in the study of Harnack inequality, the corresponding result for lower bound conditions of the curvature and the second fundamental form is still open. Nevertheless, log-Harnack and Harnack inequalities for the Neumann semigroup on non-convex manifolds have been provided by [13, Theorem 5.1] and [14, Theorem 4.1] respectively. Finally, as an extension to a result in [4] where manifolds without boundary is considered, the Lévy-Gromov isoperimetric inequality is derived in Section 4 for manifolds with boundary.

2 Gradient estimate

Let K_{1}, K_{2} ∈ C(M) be such that

Ric − ∇Z ≥ −K_{1} on M, 𝕀 ≥ −K_{2} on ∂M.  (2.1)

According to [8, Theorem 5.1] this condition implies

|∇P_{t}f| ≤ 𝔼[|∇f|(X_{t})e^{∫_{0}^{t} K_{1}(X_{s})ds+∫_{0}^{t} K_{2}(X_{s})dl_{s}}], t ≥ 0, f ∈ C^{1}(M). (2.2)

The main purpose of this section is to prove that these two statements are indeed equivalent to each other. To prove that (2.2) implies (2.1), we need the following results collected from [11, Proof of Lemma 2.1] and [13, Theorem 2.1, Lemma 2.2, Proposition A.2] respectively:

(I) For any λ > 0, 𝔼e^{λl_{t}} < ∞.

(II) For X_{0} = x ∈ ∂M, lim sup_{t→0} \frac{1}{t}|𝔼l_{t} − 2\sqrt{t/π}| < ∞.

(III) For X_{0} = x ∈ ∂M, there exists a constant c > 0 such that 𝔼l_{t}^{2} ≤ ct, t ∈ [0, 1].

(IV) Let ρ be the Riemannian distance. For δ > 0 and X_{0} = x ∈ M \ ∂M such that ρ(x, ∂M) ≥ δ, the stopping time τ_{δ} := inf{t > 0 : ρ(X_{t}, x) ≥ δ} satisfies ℙ(τ_{δ} ≤ t) ≤ c exp[−δ^{2}/(16t)] for some constant c > 0 and all t > 0.

Theorem 2.1. (2.1), (2.2) and the following inequality are equivalent to each other:

|∇P_{t}f|^{2} ≤ (P_{t}|∇f|^{2})𝔼[e^{2 ∫_{0}^{t} K_{1}(X_{s})ds+2 ∫_{0}^{t} K_{2}(X_{s})dl_{s}}], t ≥ 0, f ∈ C^{1}(M). (2.3)

Proof. Since by [8] (2.1) implies (2.2) which is stronger than (2.3) due to the Schwartz inequality, it remains to deduce (2.1) from (2.3).

(a) Proof of Ric − ∇Z ≥ −K_{1}. It suffices to prove at points in the interior. Let X_{0} = x ∈ M \ ∂M. For any ε > 0 there exists δ > 0 such that

B̄(x, δ) ⊂ M \ ∂M, sup_{y∈B̄(x,δ)} |K_{1}(y) − K_{1}(x)| ≤ ε,  (2.4)

where B̄(x, δ) is the closed geodesic ball at x with radius δ. Since l_{t} = 0 for t ≤ τ_{δ}, by (2.3), (I) and (IV) we have

\begin{matrix}|∇P_{t}f|^{2}(x) & ≤ (P_{t}|∇f|^{2}(x))𝔼e^{2 ∫_{0}^{t} K_{1}(X_{s})ds+2 ∫_{0}^{t} K_{2}(X_{s})dl_{s}} \\ ≤ (P_{t} & |∇f|^{2}(x))\{e^{2t(K_{1}(x)+ε)}ℙ(τ_{δ} ≥ t) + \sqrt{ℙ(τ_{δ} < t)𝔼e^{4t∥K_{1}∥_{∞}+4∥K_{2}∥_{∞}l_{t}}}\} \\ ≤ (P_{t} & |∇f|^{2}(x))e^{2t(K_{1}(x)+ε)} + Ce^{−λ/t}, t ∈ (0, 1] \end{matrix}

for some constants C, λ > 0. This implies

lim sup_{t→0} \frac{|∇P_{t}f|^{2}(x) − |∇f|^{2}(x)}{t} ≤ lim sup_{t→0} \frac{e^{2t(K_{1}(x)+ε)}P_{t}|∇f|^{2}(x) − |∇f|^{2}(x)}{t}.

(2.5)

Now, let f ∈ C^{∞}(M) with Nf|_{∂M} = 0, we have

P_{t}f = f + ∫_{0}^{t} P_{s}Lfds, t ≥ 0.

Then

lim sup_{t→0} \frac{|∇P_{t}f|^{2}(x) − |∇f|^{2}(x)}{t}
= lim_{t→0} \frac{1}{t}\{| ∫_{0}^{t} ∇P_{s}Lfds|^{2} + 2 ∫_{0}^{t}⟨∇f, ∇P_{s}Lf⟩ds\}(x).

(2.6)

Moreover, according to the last display in the proof of [8, Theorem 5.1] (the initial data u_{0} ∈ O_{x}(M) was missed in the right hand side therein),

∇P_{t}Lf = u_{0}𝔼[M_{t}u_{t}^{−1}∇Lf(X_{t})],

where u_{t} is the horizontal lift of X_{t} on the frame bundle O(M), and M_{t} is a d × d-matrices valued right continuous process satisfying M_{0} = I and (see [8, Corollary 3.6])

∥M_{t}∥ ≤ exp [∥K_{1}∥_{∞}t + ∥K_{2}∥_{∞}l_{t}].

So, due to (I), |∇P_{·}Lf| is bounded on [0, 1] × M and ∇P_{s}Lf → ∇Lf as s → 0. Combining this with (2.6) we obtain

lim sup_{t→0} \frac{|∇P_{t}f|^{2}(x) − |∇f|^{2}(x)}{t} = 2⟨∇f, ∇Lf⟩(x).  (2.7)

Documenta Mathematica 15 (2010) 527–543

On the other hand, applying the Itô formula to |∇f|^{2}(X_{t}) we have

\begin{aligned}P_{t}|∇f|^{2}(x) & = |∇f|^{2}(x) + ∫_{0}^{t} P_{s}L|∇f|^{2}(x)ds + 𝔼 ∫_{0}^{t} N|∇f|^{2}(X_{s})dl_{s} \\ & ≤ |∇f|^{2}(x) + ∫_{0}^{t} P_{s}L|∇f|^{2}(x)ds + ∥∇|∇f|^{2}∥_{∞}𝔼l_{t}. \end{aligned} (2.8)

Since l_{t} = 0 for t ≤ τ_{δ}, by (III) and (IV) we have

𝔼l_{t} ≤ \sqrt{(𝔼l_{t}^{2})ℙ(τ_{δ} ≤ t)} ≤ c_{1}e^{−λ/t}, t ∈ (0, 1]

for some constants c_{1}, λ > 0. So, it follows from (2.8) that

lim sup_{t→0} \frac{P_{t}|∇f|^{2}(x) − |∇f|^{2}(x)}{t} ≤ L|∇f|^{2}(x).

Combining this with (2.5) and (2.7), we arrive at

\frac{1}{2}L|∇f|^{2}(x) − ⟨∇f, ∇Lf⟩(x) ≥ −(K_{1}(x) + ε), f ∈ C^{∞}(M), Nf|_{∂M} = 0.

According to the Bochner-Weitzenböck formula, this is equivalent to (Ric − ∇Z)(x) ≥ −(K_{1}(x) + ε). Therefore, Ric − ∇Z ≥ −K_{1} holds on M by the arbitrariness of x ∈ M \ ∂M and ε > 0.

(b) Proof of 𝕀 ≥ −K_{2}. Let X_{0} = x ∈ ∂M. For any f ∈ C^{∞}(M) with Nf|_{∂M} = 0, (2.3) implies that

|∇P_{t}f|^{2}(x) ≤ e^{C_{1}t}(P_{t}|∇f|^{2}(x))𝔼e^{2 ∫_{0}^{t} K_{2}(X_{s})dl_{s}},  (2.9)

where C_{1} = 2∥K_{1}∥_{∞}. Let

ε_{t} = 2 sup_{s∈[0,t]} |K_{2}(X_{s}) − K_{2}(x)|.

By the continuity of the reflecting diffusion process we have ε_{t} ↓ 0 as t ↓ 0. Since there exists c_{0} > 0 such that for any r ≥ 0 one has e^{r} ≤ 1 + r + c_{0}r^{3/2}e^{r}, we obtain

log 𝔼e^{2 ∫_{0}^{t} K_{2}(X_{s})dl_{s}} ≤ log {1 + 2K_{2}(x)𝔼l_{t} + 𝔼(ε_{t}l_{t}) + C_{2}𝔼(l_{t}^{3/2}e^{C_{2}l_{t}})} (2.10)

for some constant C_{2} > 0. Moreover, by (I) and (III) we have

𝔼(l_{t}^{3/2}e^{C_{2}l_{t}}) ≤ (𝔼l_{t}^{2})^{3/4}(𝔼e^{4C_{2}l_{t}})^{1/4} ≤ C_{3}t^{3/4}, t ∈ (0, 1]

for some constant C_{3} > 0. Substituting this and (2.10) into (2.9), we arrive at

lim sup_{t→0} \frac{1}{\sqrt{t}} log \frac{|∇P_{t}f|^{2}(x)}{P_{t}|∇f|^{2}(x)} ≤ lim sup_{t→0} \frac{2K_{2}(x)𝔼l_{t} + 𝔼(ε_{t}l_{t})}{\sqrt{t}}.

Since 𝔼ε_{t}^{2} → 0 as t → 0 and 𝔼l_{t}^{2} ≤ ct due to (III), this and (II) imply

lim sup_{t→0} \frac{1}{\sqrt{t}} log \frac{|∇P_{t}f|^{2}(x)}{P_{t}|∇f|^{2}(x)} ≤ \frac{4K_{2}(x)}{\sqrt{π}}.

Combining this with (1.3) for p = 2 we complete the proof.

3 Proof of Theorem 1.1

Applying Theorem 2.1 to K_{1} = K and K_{2} = σ we conclude that (1), (2) and (3) are equivalent to each other. Noting that the log-Sobolev inequality (4) implies the Poincaré inequality (5) (see e.g. [6]), it suffices to prove that (2) ⇒ (4),

(5) ⇒ (1), and (2) ⇒ (6) ⇒ (7) ⇒ (1), where “ ⇒” stands for “implies”. We shall complete the proof step by step.

(a) (2) ⇒ (4). By approximations we may assume that f ∈ C^{∞}(M) with Nf|_{∂M} = 0. In this case

\frac{d}{dt}P_{t}f = LP_{t}f = P_{t}Lf.

So, for fixed t > 0 it follows from (2) that

\frac{d}{ds}P_{t−s}\{(P_{s}f^{2}) log P_{s}f^{2}\} = −P_{t−s} \frac{|∇P_{s}f^{2}|^{2}}{P_{s}f^{2}}

≥ −4e^{2Ks}P_{t−s} \frac{(𝔼[f|∇f|(X_{s})e^{σl_{s}}])^{2}}{P_{s}f^{2}}

≥ −4e^{2Ks}P_{t−s}𝔼[|∇f|^{2}(X_{s})e^{2σl_{s}}].

(3.1)

Next, by the Markov property, for ℱ_{s} = σ(X_{r} : r ≤ s), s ≥ 0, we have

P_{t−s}(𝔼[|∇f|^{2}(X_{s})e^{2σl_{s}}])(x) = 𝔼^{x}𝔼^{X_{t−s}}[|∇f|^{2}(X_{s})e^{2σl_{s}}]
= 𝔼^{x}[𝔼^{x}(e^{2σ(l_{t}−l_{t−s})}|∇f|^{2}(X_{t})|ℱ_{t−s})]
= 𝔼^{x}[|∇f|^{2}(X_{t})e^{2σ(l_{t}−l_{t−s})}].

Combining this with (3.1) we obtain

\frac{d}{ds}P_{t−s}\{(P_{s}f^{2}) log P_{s}f^{2}\} ≥ −4𝔼[|∇f|^{2}(X_{t})e^{2Ks+2σ(l_{t}−l_{t−s})}], s ∈ (0, t).

This implies (4) by integrating both sides with respect to ds from 0 to t.

(b1) (5) ⇒ Ric − ∇Z ≥ −K. Let X_{0} = x ∈ M \ ∂M and f ∈ C^{∞}(M) with Nf|_{∂M} = 0. By (5) we have

P_{t}f^{2} − (P_{t}f)^{2} ≤ 2𝔼[|∇f|^{2}(X_{t}) ∫_{0}^{t} e^{2Ks+2σ(l_{t}−l_{t−s})}ds].  (3.2)

Let δ > 0 and τ_{δ} be as in the proof of Theorem 2.1(a). Then

𝔼[|∇f|^{2}(X_{t}) ∫_{0}^{t} e^{2Ks+2σ(l_{t}−l_{t−s})}ds]

≤ (P_{t}|∇f|^{2}) ∫_{0}^{t} e^{2Ks}ds + t∥∇f∥_{∞}e^{2Kt}𝔼[e^{2σl_{t}}1_{{τ_{δ}<t}}]

≤ \frac{e^{2Kt} − 1}{2K}P_{t}|∇f|^{2}(x) + ce^{−λ/t}, t ∈ (0, 1]

holds for some constants c, λ > 0 according to (IV). Combining this with (3.2) we conclude that

P_{t}f^{2}(x) − (P_{t}f)^{2}(x) ≤ \frac{e^{2Kt} − 1}{K}P_{t}|∇f|^{2}(x) + 2ce^{−λ/t}, t ∈ (0, 1]. (3.3)

Since f ∈ C^{∞}(M) with Nf|_{∂M=0}, we have

\begin{aligned}P_{t}f^{2} − (P_{t}f)^{2} & = f^{2} + ∫_{0}^{t} P_{s}Lf^{2}ds − (f + ∫_{0}^{t} P_{s}Lfds)^{2} \\ & = ∫_{0}^{t}(P_{s}Lf^{2} − 2fP_{s}Lf)ds − ( ∫_{0}^{t} P_{s}Lfds)^{2}. \end{aligned}  (3.4)

Moreover, by the continuity of s ↦ P_{s}Lf, we have

( ∫_{0}^{t} P_{s}Lfds)^{2} = (Lf)^{2}t^{2} + ∘(t^{2}),  (3.5)

where and in what follows, for a positive function (0, 1] ∋ t ↦ ξ_{t} the notion ∘(ξ_{t}) stands for a variable such that ∘(ξ_{t})/ξ_{t} → 0 as t → 0; while ◯(ξ_{t}) satisfies that ◯(ξ_{t})/ξ_{t} is bounded for t ∈ (0, 1]. Moreover, since

\begin{matrix}P_{s}Lf^{2} − 2fP_{s}Lf = & Lf^{2} − 2fLf + ∫_{0}^{s}(P_{r}L^{2}f^{2} − 2fP_{r}L^{2}f)dr \\ + 𝔼 ∫_{0}^{s}(NLf^{2} − 2f(x)NLf)(X_{r})dl_{r}, \end{matrix}

and due to (IV)

|𝔼 ∫_{0}^{t} {NLf^{2} − 2f(x)NLf}(X_{r})dl_{r}| ≤ c_{1}𝔼l_{s} ≤ c_{2}e^{−λ/s}, s ∈ (0, 1]

holds for some constants c_{1}, c_{2}, λ > 0, it follows from the continuity of P_{s} in s that

∫_{0}^{t}(P_{s}Lf^{2} − 2fP_{s}Lf)ds = 2t|∇f|^{2} + \frac{t^{2}}{2} (L^{2}f^{2} − 2fL^{2}f) + ∘(t^{2}).

Combining this with (3.4) and (3.5) we obtain

\begin{matrix}P_{t}f^{2}(x)− & (P_{t}f)^{2}(x) = \\ = 2t|∇f|^{2}(x) + \frac{t^{2}}{2} (L^{2}f^{2} − 2fL^{2}f)(x) − t^{2}(Lf)^{2}(x) + ∘(t^{2}) \\ = 2t|∇f|^{2}(x) + t^{2}(2⟨∇f, ∇Lf⟩ + L|∇f|^{2})(x) + ∘(t^{2}). \end{matrix} (3.6)

Similarly,

\begin{aligned}P_{t}|∇f|^{2}(x) & = |∇f|^{2}(x) + ∫_{0}^{t} P_{s}L|∇f|^{2}(x)ds + 𝔼 ∫_{0}^{t} N|∇f|^{2}(X_{s})dl_{s} \\ & = |∇f|^{2}(x) + tL|∇f|^{2}(x) + ∘(t). \end{aligned}

Combining this with (3.3) and (3.6) we arrive at

\begin{matrix}\frac{1}{t^{2}} \{t^{2}(2⟨∇f, ∇Lf⟩ + L|∇f|^{2})(x) + ∘(t^{2})\} \\ ≤ \frac{e^{2Kt} − 1}{Kt}L|∇f|^{2}(x) + ∘(1) + \frac{1}{t}(\frac{e^{2Kt} − 1}{Kt} − 2)|∇f|^{2}(x). \end{matrix}

Letting t → 0 we obtain

L|∇f|^{2}(x) − 2⟨∇f, ∇Lf⟩(x) ≥ −2K|∇f|^{2}(x),

which implies (Ric − ∇Z)(x) ≥ −K by the Bochner-Weitzenböck formula. (b2) (5) ⇒ 𝕀 ≥ −σ. Let X_{0} = x ∈ ∂M and f ∈ C^{∞}(M) with Nf|_{∂M} = 0. Noting that Lf^{2} − 2fLf = 2|∇f|^{2}, by the Itô formula we have

\begin{matrix}P_{t}f^{2}(x) − (P_{t}f)^{2}(x) = f^{2} + ∫_{0}^{t} P_{s}Lf^{2}ds − (f + ∫_{0}^{t} P_{s}Lfds)^{2} \\ = 2 ∫_{0}^{t} P_{s}|∇f|^{2}(x)ds + 2 ∫_{0}^{t}[P_{s}(fLf)(x) − f(x)P_{s}Lf(x)]ds + ◯(t^{2}). \end{matrix} (3.7)

Since Nf|_{∂M} = 0 implies

0 = ⟨∇f, ∇⟨N, ∇f⟩⟩ = Hess_{f}(N, ∇f) − 𝕀(∇f, ∇f),

it follows that

𝕀(∇f, ∇f) = Hess_{f}(N, ∇f) = \frac{1}{2}N|∇f|^{2}.  (3.8)

So, by the Itô formula, (II) and (III) yield

\begin{aligned}P_{s}|∇f|^{2}(x) & = |∇f|^{2}(x) + ∫_{0}^{s} P_{r}L|∇f|^{2}(x)dr + 𝔼 ∫_{0}^{s} N|∇f|^{2}(X_{r})dl_{r} \\ & = |∇f|^{2}(x) + ◯(s) + 2𝔼 ∫_{0}^{s} 𝕀(∇f, ∇f)(X_{r})dl_{r} \\ & = |∇f|^{2}(x) + \frac{4\sqrt{s}}{\sqrt{π}}𝕀(∇f, ∇f)(x) + ∘(s^{1/2}). \end{aligned} (3.9)

Moreover, since (fNLf)(X_{r}) − f(x)(NLf)(X_{r}) is bounded and goes to zero as r → 0, it follows from (III) that

2𝔼 ∫_{0}^{t} ds ∫_{0}^{s}[(fNf)(X_{r}) − f(x)(NLf)(X_{r})]dl_{r} = ∘(t^{3/2}).

So, by the Iô formula

2 ∫_{0}^{t}[P_{s}(fLf)(x) − f(x)P_{s}Lf(x)]ds
= 2 ∫_{0}^{t} ds ∫_{0}^{s}[P_{r}L(fLf)(x) − f(x)P_{r}L^{2}f(x)]dr
+ 2𝔼 ∫_{0}^{t} ds ∫_{0}^{s}[(fNLf)(X_{r}) − f(x)(NLf)(X_{r})]dl_{r} = ∘(t^{3/2}).

Combining this with (3.7) and (3.9) we arrive at

\begin{matrix}lim_{t→0} \frac{1}{t\sqrt{t}}(P_{t}f^{2}(x) − (P_{t}f)^{2}(x) − 2t|∇f|^{2}(x)) \\ = \frac{8}{\sqrt{π}} 𝕀(∇f, ∇f)(x) lim_{t→0} \frac{1}{t\sqrt{t}} ∫_{0}^{t} \sqrt{s} ds = \frac{16}{3\sqrt{π}} 𝕀(∇f, ∇f)(x). \end{matrix}  (3.10)

On the other hand, by the Itô formula for |∇f|^{2}(X_{t}), it follows from (3.8) and (II) that A_{t} :=

= \frac{1}{t\sqrt{t}}𝔼\{|∇f|^{2}(X_{t}) ∫_{0}^{t} e^{2Ks+2σ(l_{t}−l_{t−s})}ds − t|∇f|^{2}(x)\}

= \frac{1}{\sqrt{t}}(𝔼|∇f|^{2}(X_{t}) − |∇f|^{2}(x)) + 𝔼\{\frac{|∇f|^{2}(X_{t})}{t\sqrt{t}} ∫_{0}^{t} (e^{2Ks+2σ(l_{t}−l_{t−s})} − 1)ds\}

= \frac{1}{\sqrt{t}}\{ ∫_{0}^{t} P_{s}L|∇f|^{2}(x)ds + 𝔼 ∫_{0}^{t} N|∇f|^{2}(X_{s})dl_{s}\}

+ 𝔼\{\frac{|∇f|^{2}(X_{t})}{t\sqrt{t}} ∫_{0}^{t} (e^{2Ks+2σ(l_{t}−l_{t−s})} − 1)ds\}

= \frac{4}{\sqrt{π}} 𝕀(∇f, ∇f)(x) + ∘(1) + 𝔼\{\frac{|∇f|^{2}(X_{t})}{t\sqrt{t}} ∫_{0}^{t} (e^{2Ks+2σ(l_{t}−l_{t−s})} − 1)ds\}.

(3.11)

Since by (I) and (III)

|𝔼[(|∇f|^{2}(X_{t}) − |∇f|^{2}(x)) ∫_{0}^{t} (e^{2Ks+2σ(l_{t}−l_{t−s})} − 1)ds]|
≤ t{𝔼(|∇f|^{2}(X_{t}) − |∇f|^{2}(x))^{2}}^{1/2}{𝔼(e^{2Kt+2σl_{t}} − 1)^{2}}^{1/2}
= ∘(t) · (𝔼[4σ^{2}l_{t}^{2}] + ∘(t)) = ∘(t^{2}),

it follows from (I) and (II) that

𝔼[|∇f|^{2}(X_{t}) ∫_{0}^{t} (e^{2Ks+2σ(l_{t}−l_{t−s})} − 1)ds]

= ∘(t^{2}) + |∇f|^{2}(x)𝔼 ∫_{0}^{t} (e^{2Ks+2σ(l_{t}−l_{t−s})} − 1)ds

= ∘(t^{3/2}) + \frac{4σ|∇f|^{2}(x)}{\sqrt{π}} ∫_{0}^{t} (\sqrt{t} − \sqrt{t − s})ds

= \frac{4σt\sqrt{t}}{3\sqrt{π}}|∇f|^{2}(x) + ∘(t^{3/2}).

Combining this with (3.11) we arrive at

A_{t} ≤ ∘(1) + \frac{4}{\sqrt{π}} 𝕀(∇f, ∇f)(x) + \frac{4σ}{3\sqrt{π}} |∇f|^{2}(x).

So, (3.10) and (5) imply that

\frac{16}{3\sqrt{π}} 𝕀(∇f, ∇f)(x) ≤ lim sup_{t→0} 2A_{t} ≤ \frac{8}{\sqrt{π}} 𝕀(∇f, ∇f)(x) + \frac{8σ}{3\sqrt{π}} |∇f|^{2}(x).

Therefore, 𝕀(∇f, ∇f)(x) ≥ −σ|∇f|^{2}(x).

(c) (2) ⇒ (6). Let f ≥ 0 be smooth satisfying the Neumann boundary condition. We have

\frac{d}{ds}P_{s}\{(P_{t−s}f) log P_{t−s}f\} = P_{s} \frac{|∇P_{t−s}f|^{2}}{P_{t−s}f} .

This implies

P_{t}(f log f) − (P_{t}f) log P_{t}f = ∫_{0}^{t} P_{s} \frac{|∇P_{t−s}f|^{2}}{P_{t−s}f} ds.  (3.12)

On the other hand, by (2) and applying the Schwartz inequality to the probability measure \frac{2K}{1−exp[−2Kt]}e^{−2Ks}ds on [0, t], we obtain

|∇P_{t}f|^{2} =

= \{\frac{2K}{1 − e^{−2Kt}} ∫_{0}^{t} |∇P_{s}(P_{t−s}f)|e^{−2Ks}ds\}^{2}

≤ \{\frac{2K}{1 − e^{−2Kt}} ∫_{0}^{t} E[|∇P_{t−s}f|(X_{s})e^{σl_{s}−Ks}]ds\}^{2}

≤ (\frac{2K}{1 − e^{−2Kt}} )^{2}(𝔼 ∫_{0}^{t} \frac{|∇P_{t−s}f|^{2}}{P_{t−s}f} (X_{s})ds) ∫_{0}^{t} 𝔼[P_{t−s}f(X_{s})e^{2σl_{s}−2Ks}]ds

= (\frac{2K}{1 − e^{−2Kt}} )^{2}( ∫_{0}^{t} P_{s} \frac{|∇P_{t−s}f|^{2}}{P_{t−s}f} ds) ∫_{0}^{t} 𝔼[P_{t−s}f(X_{s})e^{2σl_{s}−2Ks}]ds.

Combining this with (3.12) and noting that the Markov property implies

\begin{aligned}𝔼[P_{t−s}f(X_{s})e^{2σl_{s}}] & = 𝔼[(𝔼^{X_{s}}f(X_{t−s}))e^{2σl_{s}}] = 𝔼[e^{2σl_{s}}𝔼(f(X_{t})|ℱ_{s})] \\ & = 𝔼[𝔼(f(X_{t})e^{2σl_{s}}|ℱ_{s})] = 𝔼[f(X_{t})e^{2σl_{s}}], \end{aligned}

we obtain (6).

(d) (6) ⇒ (7). The proof is similar to the classical one for the log-Sobolev inequality to imply the Poincaré inequality. Let f ∈ C^{∞}(M). SInce M is compact, 1 + εf > 0 for small ε > 0. Applying (6) to 1 + εf in place of f, we obtain

|∇P_{t}f|^{2} ≤\frac{2K}{ε^{2}(1 − e^{−2Kt})}\{P_{t}(1 + εf) log(1 + εf) − (1 + εP_{t}f) log(1 + εP_{t}f)\}
· 𝔼{(1 + εf(X_{t})) ∫_{0}^{t} e^{2σl_{s}−2Ks}ds}.

(3.13)

Since by Taylor’s expansion

P_{t}(1 + εf) log(1 + εf) − (1 + εP_{t}f) log(1 + εP_{t}f) = \frac{ε^{2}}{2} (P_{t}f^{2} − (P_{t}f)^{2}) + ∘(ε^{2}),

letting ε → 0 in (3.13) we obtain (7). (e1) (7) ⇒ Ric − ∇Z ≥ −K. Let X_{0} = x ∈ M \ ∂M and f ∈ C^{∞}(M) with Nf|_{∂M} = 0. by (I) and (IV) we have

𝔼e^{2σl_{s}} = 1 + 𝔼[e^{2σl_{s}}1_{{τ_{δ}≤s}}] = 1 + ∘(s).

So,

𝔼 ∫_{0}^{t} e^{2σl_{s}−2Ks}ds = \frac{1 − exp[−2Kt]}{2K} + ∘(t).

Combining this with (3.6) and (7), we conclude that, at point x,

\frac{|∇P_{t}f|^{2} − |∇f|^{2}}{t} ≤

≤ \frac{K}{1 − e^{−2Kt}} \{2|∇f|^{2} + t(2⟨∇f, ∇Lf⟩ + L|∇f|^{2})\} − \frac{|∇f|^{2}}{t} + ∘(1)

= \frac{1}{t}(\frac{2Kt}{1 − e^{−2Kt}} − 1)|∇f|^{2} + \frac{Kt}{1 − e^{−2Kt}} (2⟨∇f, ∇Lf⟩ + L|∇f|^{2}) + ∘(1).

Letting t → 0 and using (2.7), we obtain

2⟨∇f, ∇Lf⟩ ≤ K|∇f|^{2} + ⟨∇f, ∇Lf⟩ + \frac{1}{2}L|∇f|^{2}

at point x. This implies Ric − ∇Z ≥ −K at this point according to the Bochner-Weitzenböck formula. (e2) (7) ⇒ 𝕀 ≥ −σ. Let X_{0} = x ∈ ∂M and f ∈ C^{∞}(M) with Nf|_{∂M} = 0. It follows from (3.10), (7) and (II) that at point x,

|∇P_{t}f|^{2} ≤

≤ \frac{2K^{2}}{(1 − e^{−2Kt})^{2}} (2t|∇f|^{2} + \frac{16t^{3/2}}{3\sqrt{π}} 𝕀(∇f, ∇f) + ∘(t^{3/2}))(t + \frac{8σt^{3/2}}{3\sqrt{π}} + ∘(t^{3/2}))

= \frac{4K^{2}t^{2}}{(1 − e^{−2Kt})^{2}} |∇f|^{2} + \frac{4K^{2}t^{5/2}}{(1 − e^{−2Kt})^{2}} ( \frac{8}{3\sqrt{π}} 𝕀(∇f, ∇f) + \frac{8σ}{3\sqrt{π}} |∇f|^{2}) + ∘(t^{1/2}).

Combining this with (2.7) we deduce at point x that

0 = lim_{t→0} \frac{1}{\sqrt{t}}(|∇P_{t}f|^{2} − \frac{4K^{2}t^{2}}{(1 − e^{−2Kt})^{2}} |∇f|^{2})

≤ lim_{t→0} \frac{4K^{2}t^{2}}{(1 − e^{−2Kt})^{2}} (\frac{8}{3\sqrt{π}} 𝕀(∇f, ∇f) + \frac{8σ}{3\sqrt{π}} |∇f|^{2})

= \frac{8}{3\sqrt{π}} 𝕀(∇f, ∇f) + \frac{8σ}{3\sqrt{π}} |∇f|^{2}.

Therefore, 𝕀(∇f, ∇f)(x) ≥ −σ|∇f|^{2}(x).

4 Lévy-Gromov isoperimetric inequality

As a dimension-free version of the classical Lévy-Gromov isoperimetric inequality, it is proved in [4] that if M does not have boundary then for V ∈ C^{2}(M) such that Ric − Hess_{V} ≥ R > 0 the following inequality

𝒰 (μ(f)) ≤ ∫_{M} \sqrt{𝒰^{2}(f) + R^{−1}|∇f|^{2}} dμ,  (4.1)

holds for any smooth function f with values in [0, 1], where μ(dx) := C(V)^{−1}e^{V (x)}dx for C(V) = ∫_{M} e^{V (x)}dx is a probability measure on M, and 𝒰 = φ ∘ Φ^{−1} for Φ(r) = (2π)^{−1} ∫_{−∞}^{r} e^{−s^{2}/2}ds and φ = Φ′. Since 𝒰 (0) = 𝒰 (1) = 0, taking f = 1_{A} (by approximations) in (4.1) for a smooth domain A ⊂ M, we obtain the isoperimetric inequality

R𝒰 (A) ≤ μ_{∂}(∂A),  (4.2)

where μ_{∂}(∂A) is the area of ∂A induced by μ. This inequality is crucial in the study of Gaussian type concentration of μ (see [4, 9]). Obviously, (4.1) follows from the following semigroup inequality by letting t → ∞:

𝒰 (P_{t}f) ≤ P_{t}\sqrt{𝒰^{2}(f) + R^{−1}(1 − e^{−2Rt})|∇f|^{2}} .  (4.3)

In this section we aim to extend (4.3) to manifolds with boundary. Now, let again M be compact with boundary ∂M, and let P_{t} be the Neumann semigroup generated by L = Δ + Z. We shall prove an analogue of (4.3) for the curvature and second fundamental condition in Theorem 1.1(1).

Theorem 4.1. Let Ric − ∇Z ≥ −K and 𝕀 ≥ −σ for some constants K ∈ ℝ and σ ≥ 0. Then for any smooth function f with values in [0, 1],

𝒰 (P_{t}f) ≤ 𝔼\sqrt{𝒰^{2}(f)(X_{t}) + |∇f|^{2}(X_{t})\frac{(e^{2Kt} − 1)e^{2σl_{t}}}{K}} , t ≥ 0.  (4.4)

If in particular ∂M is convex (i.e. σ = 0), then

𝒰 (P_{t}f) ≤ P_{t}\sqrt{𝒰^{2}(f) + |∇f|^{2}(X_{t})\frac{e^{2Kt} − 1}{K}} , t ≥ 0.

If moreover K < 0, then (4.1) and (4.2) hold for R = −K > 0.

Proof. It suffices to prove the first assertion. To this end, we shall use the following equivalent condition for Ric − ∇Z ≥ −K (see e.g. the proof of [9, (1.14)]):

Γ_{2}(f, f) := \frac{1}{2}L|∇f|^{2} − ⟨∇f, ∇Lf⟩ ≥ −K|∇f|^{2} + \frac{|∇|∇f|^{2}|^{2}}{4|∇f|^{2}} .  (4.5)

Semigroup Properties for the Second Fundamental Form 541 To prove (4.4), we consider the process

η_{s} = 𝒰^{2}(P_{t−s}f)(X_{s}) + |∇P_{t−s}f|^{2}(X_{s})\frac{(e^{2Ks} − 1)e^{2σl_{s}}}{K} , s ∈ [0, t].

To apply the Itô formula for η_{s}, recall that X_{s} solves the equation

dX_{s} = \sqrt{2} u_{s} ∘ dB_{s} + N(X_{s})dl_{s},

where u_{s} is the horizontal lift of X_{s} and B_{s} is the Brownian motion on ℝ^{d} provided M is d-dimensional. So,

dη_{s} = \sqrt{2}⟨2(𝒰 𝒰′)(P_{t−s}f)(X_{s}) + \frac{(e^{2Ks} − 1)e^{2σl_{s}}}{K} ∇|∇P_{t−s}f|^{2}(X_{s}), u_{s}dB_{s}⟩

+ \{2(𝒰′ ^{2} + 𝒰 𝒰′′)(P_{t−s}f)|∇P_{t−s}f|^{2} + 2Γ_{2}(P_{t−s}f, P_{t−s}f)\frac{(e^{2Ks} − 1)e^{2σl_{s}}}{K}

+ 2|∇P_{t−s}f|^{2}e^{2Ks+2σl_{s}}}(X_{s})ds

+ \frac{(e^{2Ks} − 1)e^{2σl_{s}}}{K} (N|∇P_{t−s}f|^{2} + 2σ|∇P_{t−s}f|^{2})(X_{s})dl_{s}.

Noting that 𝒰 𝒰′′ = −1 and σ ≥ 0 so that e^{2σl_{s}} ≥ 1, combining this with (3.8), 𝕀 ≥ −σ and (4.5), we obtain

\begin{aligned}dη_{s} & ≥ \sqrt{2}⟨2(𝒰 𝒰′)(P_{t−s}f)(X_{s}) + \frac{(e^{2Ks} − 1)e^{2σl_{s}}}{K} ∇|∇P_{t−s}f|^{2}(X_{s}), u_{s}dB_{s}⟩ \\ & + \{2𝒰′ ^{2}(P_{t−s}f)|∇P_{t−s}f|^{2} + \frac{(e^{2Ks} − 1)e^{2σl_{s}}|∇|∇P_{t−s}f|^{2}|^{2}}{2K|∇P_{t−s}f|^{2}} \}(X_{s})ds. \end{aligned}

Therefore, there exists a martingale M_{s} for s ∈ [0, t] such that

dη_{s}^{1/2} = dM_{s} + \frac{dη_{s}}{2η_{s}^{1/2}} −

− \frac{|2(𝒰 𝒰′)(P_{t−s}f)∇P_{t−s}f + \frac{(e^{2Ks}−1)e^{2σls}}{K} ∇|∇P_{t−s}f|^{2}|^{2}(X_{s})}{4η_{s}^{3/2}}

= dM_{s} + \frac{1}{4η_{s}^{3/2}} B_{s}ds,

where

Documenta Mathematica 15 (2010) 527–543

B_{s} :=2η_{s}(2𝒰′ ^{2}(P_{t−s}f)|∇P_{t−s}f|^{2} + \frac{(e^{2Ks} − 1)e^{2σl_{s}}|∇|∇P_{t−s}f|^{2}|^{2}}{2K|∇P_{t−s}f|^{2}} )(X_{s})

− |2(𝒰 𝒰′)(P_{t−s}f)∇P_{t−s}f + \frac{e^{2Ks} − 1}{K}e^{2σl_{s}}∇|∇P_{t−s}f|^{2}|^{2}(X_{s})

≥\frac{(e^{2Ks} − 1)e^{2σl_{s}}}{K} \{\frac{𝒰^{2}(P_{t−s}f)|∇|∇P_{t−s}f|^{2}|^{2}}{2|∇P_{t−s}f|^{2}} + 4|∇P_{t−s}f|^{4}𝒰′ ^{2}(P_{t−s}f)
− 4(𝒰 𝒰′)(P_{t−s}f)⟨∇P_{t−s}f, ∇|∇P_{t−s}f|^{2}⟩}(X_{s})
≥0.

So, η_{s}^{1/2} is a sub-martingale on [0, t]. Therefore, 𝔼η_{0}^{1/2} ≤ 𝔼η_{t}^{1/2}, which is nothing but (4.4).

References

[1] D. Bakry, Transformations de Riesz pour les semigroupes symétriques, Lecture Notes in Math. No. 1123, 130–174, Springer, 1985.

[2] D. Bakry, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, New Trends in Stochastic Analysis, 43–75, World Scientific, 1997.

[3] D. Bakry and M. Emery, Hypercontractivité de semi-groupes de diffusion, C. R. Acad. Sci. Paris. Sér. I Math. 299(1984), 775–778.

[4] D. Bakry and M. Ledoux, Lévy-Gromov’s isoperimetric inequality for an infinite dimensional diffusion operator, Invent. Math. 123(1996), 259–281.

[5] H. Donnely and P. Li, Lower bounds for the eigenvalues of Riemannian manifolds, Michigan Math. J. 29 (1982), 149–161.

[6] J.-D. Deuschel and D. W. Stroock, Large Deviations, Academic Press, New York, 1989.

[7] K. D. Elworthy, Stochastic flows on Riemannian manifolds, Diffusion Processes and Related Problems in Analysis, vol. II, Progress in Probability, 27, 37–72, Birkhäuser, 1992.

[8] E. P. Hsu, Multiplicative functional for the heat equation on manifolds with boundary, Michigan Math. J. 50(2002), 351–367.

[9] M. Ledoux, The geometry of Markov diffusion generators, Ann. de la Facul. des Sci. de Toulouse 9 (2000), 305–366.

[10] F.-Y. Wang, Equivalence of dimension-free Harnack inequality and curvature condition, Int. Equ. Operat. Theory 48(2004), 547–552.

[11] F.-Y. Wang, Gradient estimates and the first Neumann eigenvalue on manifolds with boundary, Stoch. Proc. Appl. 115(2005), 1475–1486.

[12] F.-Y. Wang, Second fundamental form and gradient of Neumann semigroups, J. Funct. Anal. 256(2009), 3461–3469.

[13] F.-Y. Wang, Harnack inequalities on manifolds with boundary and applications, to appear in J. Math. Pures Appl.

[14] F.-Y. Wang, Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on non-convex manifolds, to appear in Annals of Probab.

Feng-Yu Wang School of Mathematical Sci. and Lab. Math. Com. Sys., Beijing Normal University Beijing 100875 China wangfy@bnu.edu.cn Alternate Address: Department of Mathematics Swansea University Singleton Park SA2 8PP UK F.Y.Wang@swansea.ac.uk 544

Documenta Mathematica 15 (2010)