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The weighted Laplacians on real and complex metric measure spaces

In this short note we compare the weighted Laplacians on real and complex (K\"ahler) metric measure spaces. In the compact case K\"ahler metric measure spaces are considered on Fano manifolds for the study of K\"ahler-Einstein metrics while real metric measure spaces are considered with Bakry-\'Emery Ricci tensor. There are twisted Laplacians which are useful in both cases but look alike each other. We see that if we consider noncompact complete manifolds significant differences appear.


Introduction
The weighted Laplacians can be considered on smooth metric measure spaces using the exterior derivative d on Riemannian manifolds and the ∂-operator on complex Kähler manifolds. We considered in [4] and [5] the weighted ∂-Laplacian also represents 2πc 1 (M ). We say λ is an eigenvalue of ∆ F if ∆ F u + λu = 0 for some nonzero complex-valued function u. If we use the weighted volume dµ = e F ω m we have where g(∂u, ∂v) = ∇ j u∇ j v = g ij ∂u ∂z j ∂v ∂z i .
Thus all the eigenvalues are nonnegative real numbers. It is shown that the first non-zero eigenvalue λ 1 (∆ F ) satisfies (1) λ 1 (∆ F ) ≥ 1 and the equality holds if and only if the Lie algebra h(M ) of all holomorphic vector fields is non-zero. In fact, given a holomorphic vector field X, there corresponds an eigenfunction u by X = grad ′ u := g ij ∂u ∂z j ∂ ∂z i . This observation was made before the publication of [3], and was inspired by Shoshichi Kobayashi's book [15], Chapter III, section 7, "Conformal changes of the Laplacian". In fact the obstruction to the existence of Kähler-Einstein metrics introduced in [3] was found using the above observation combined with an idea suggested by J.L.Kazdan [13] (see also [14]). See section 2 for more about the application of the above observation to the study of Kähler-Einstein metrics. The proof of (1) follows from the general formula [21], [23], [16] for other applications to the study in Kähler geometry.
On the other hand a similar idea is commonly used in Riemannian geometry and probability theory with the Bakry-Émery Ricci curvature Ric +∇ 2 f on the weighted Riemannian manifolds (M, g, f ). This means that (M, g) is a complete Riemannian manifold with the weighted measure dµ = e −f dv, where dv denotes the Riemannian volume measure on (M, g) and f is a real-valued C 2 -function. We denote by C ∞ (M ) (resp. C ∞ 0 (M )) the set of real-valued smooth functions (reps. with compact support). For all u, v ∈ C ∞ 0 (M ), the following integration by parts formula holds where ∆ f is the called the weighted Laplacian with respect to the volume measure µ. More precisely, we have Here we denote by ∆ the d-Laplacian: ∆ = −d * d = g ij ∇ i ∇ j with respect to real coordinates (x 1 , · · · , x n ). In [1], Bakry andÉmery proved that for all u ∈ C ∞ 0 (M ), The formula (3) can be viewed as a natural extension of the Bochner-Weitzenböck formula. The equation (2) can also be derived from a similar Weitzeböck type formula. The probabilistic study of the weighted Laplacian was motivated by the hypercontractivity of Markov semigroups. In [8] Gross showed that the hypercontractivity holds if and only if the logarithmic Sobolev inequality holds. Then in [1] Bakry andÉmery showed that on a smooth metric measure space the logarithmic Sobolev inequality holds if the Bakry-Émery Ricci tensor is bounded from below by a positive constant, that is if there is a positive constant C such that (4) Ric +∇ 2 f ≥ Cg.
Note in this case we have See section 3 more about the Bakry-Émery Ricci tensor. It has been a puzzle (to the author) how the real and complex (Fano) cases are different. One immediate difference is that, while ∆ f in the Riemannian case is a real operator, ∆ F in the Kähler case is not a real operator unless F is constant. This means that ∆ F u is a complex valued function even if u is real valued. Therefore the eigenfunctions corresponding to nonzero eigenvalues can not be real valued.
In this paper we see, by comparing with the results of Cheng and Zhou [2] in the real noncompact case, that if we consider noncompact complete manifolds then more significant differences appear between the real and complex weighted Laplacians. For example, when the first nonzero eigenvalue of the twisted Laplacian attains the expected lower bound, it is a discrete spectrum in the real case, but it can be an essential spectrum in the complex case. Moreover if the the first nonzero eigenvalue of expected lower bound has multiplicity k then in the real case the Gaussian soliton of dimension k splits off, but this is not the case in the complex case. See sections 4 and 5 for more detail.

The case of Fano manifolds
Let M have positive first Chern class, i.e. the first Chern class c 1 (M ) contains a positive closed (1, 1)-form. This is equivalent to say the anticanonical bundle of M is ample, and such a manifold is called a Fano manifold. We choose a Kähler from Since the Ricci form also represents 2πc 1 (M ) there exists a smooth function F such that Denote by C ∞ C (M ) the set of all complex valued functions on M . For u and v in C ∞ C (M ), we consider the L 2 -inner product with respect to the weighted volume We considered in [4] and [5] the weighted ∂-Laplacian We then have We say λ is an eigenvalue of ∆ F if ∆ F u + λu = 0 for some nonzero complex-valued function u. Since ∆ F is self-adjoint by (10) all the eigenvalues are nonnegative real numbers.
Theorem 2.1 (c.f. [4], [5]). The first non-zero eigenvalue λ 1 (∆ F ) of ∆ F satisfies (11) λ 1 (∆ F ) ≥ 1 and the equality holds if and only if the Lie algebra h(M ) of all holomorphic vector fields is non-zero. In fact, for a holomorphic vector field X, there corresponds an eigenfunction u by If we pick another Kähler form ω = ω + √ −1∂∂ϕ with ϕ ∈ C ∞ (M ), theñ u := u + u i ϕ i is the first eigenfunction corresponding to the holomorphic vector field X in the previous theorem. Using this I proved the following theorem in the first version of [3]. If we use the above correspondence in Theorem 2.1 we have a complex valued smooth function u such that X = grad ′ u and that ∆ F u + u = 0. It follows that XF = −∆ ∂ u − u and that (12) f This implies the following result which is known as Mabuchi's theorem [17] for toric Fano manifolds. But this is generally true even if M is not toric. Note also in passing that the theorem of Matsushima [19], saying that the Lie algebra h(M ) is reductive when M is a Kähler-Einstein manifold, follows from the Theorem 2.1 since if (M, g) is Kähler-Einsten then we can take F = 0 and ∆ F = ∆ ∂ is a real operator so that h(M ) is a complexification of the purely imaginary eigenfunctions, the gradient vectors of which are Killing vector fields.

Hypercontractivity, logarithmic Sobolev inequality and Bakry-Émery-Ricci tensor
In this section we review the historical background of Bakry-Émery Ricci tensor in probability theory. We refer the reader to the lecture notes [11] for more details. A Markov semigroup P t = e tL , t > 0, is said to have hypercontractivity if ∞) is a constant and µ is a probability measure.
In [8], L.Gross showed that P t = e tL satisfies hypercontractivity if and only if the following logarithmic Sobolev inequality is satisfied: where c ∈ (0, ∞) is a constant independent of u.
Let (M, g, e −f dV g ) be a smooth metric measure space. This means that (M, g) is a Riemannian manifold with the Riemannian volume element dV g , that f is a smooth function on M , and that we consider the twisted volume element e −f dV g . In [1], Bakry andÉmery showed that if a metric measure space (M, g, e −f dV g ) has finite measure M e −f dV g < ∞, and Ric f := Ric + ∇ 2 f ≥ λg with a positive constant λ > 0, then the logarithmic Sobolev inequality holds with respect to the measure dµ = e −f dV g and Lu = ∆ f u = ∆u − ∇ i f ∇ i u. In particular, P t = e tL satisfies hypercontractivity. Note that Morgan [20] later proved that if Ric f ≥ λg for some positive constant λ > 0 then the weighted measure is finite: M e −f dV g < ∞.
The key observation of Bakry andÉmery is the following Weitzenböck type formula.
As an immediate consequence of (13) we obtain the following spectral gap.
It is known (c.f. [12], [2]) that if the logarithmic Sobolev inequality holds then we have the compact embedding H 1 f (M ) ֒→ L 2 f (M ), and the spectrum of the twisted Laplacian ∆ f is discrete. We denote by λ 1 (∆ f ) the first nonzero eigenvalue of ∆ f with eigenfunction in H 1 f (M ). Now we recall a recent result of Cheng and Zhou [2]. First let us see two examples.
Take f = |t| 2 4 with t ∈ R k . Then we have Ric f = 1 2 g and λ 1 (∆ f ) = 1 2 , and the corresponding eigenfunctions are linear functions in R k .

Theorem 4.3 ([2]
). Let (M n , g, e −f dV g ) be a complete smooth metric measure space with Ric f ≥ λg for a positive constant λ. If λ 1 (∆ f ) = λ with multiplicity k then M is isometric to Σ n−k × R k with λ 1 (∆ Σ f ) > λ and f is written in the form f (p, t) = f (p, 0) + λ 2 |t| 2 where p ∈ Σ and t ∈ R k . In the case of shrinking Ricci soliton (14) Ric . This is because we always have and f belongs to H 1 f (M ). In the case of a compact shrinking soliton, a lower bound of λ 1 (∆ f ) can be given in terms of the diameter, and combined with (15) we can obtain a universal lower bound of the diameter ( [7], [6]).

Complete Kähler metric measure spaces
We say that (M, g, e F dV g ) is a complete Kähler metric measure space if (M, g) is a complete Kähler manifold with (16) Ric is the Kähler form and F is a smooth function. If M is compact M is naturally a Fano manifold. Note also (M, g) is a gradient Kähler-Ricci soliton if in addition ∇ ′′ ∇ ′′ F = 0.
By the same arguments of Morgan [20] we can show that the weighted volume M e F dV g is finite. We consider the same weighted Laplacian ∆ F as in the Fano case in section 2. Namely, ∆ F is given by the same formula (9) and acts on the complex-valued functions C ∞ C (M ). Theorem 5.1. Let (M, g, e F dV g ) be a complete Kähler metric measure space. Then λ 1 (∆ F ) ≥ 1 and there is an imbedding of the 1-eigenspace Λ 1 to the Lie algebra h(M ) of all holomorphic vector fields on M .
Unlike the real case in section 3, Λ 1 can be infinite dimensional so that 1 is an essential spectrum. For example, consider (C n , g can , e −|z| 2 dV gcan ) with n ≥ 2 where g can = ∇z ·∇z. With F = −|z| 2 , (C n , g can , e −|z| 2 dV gcan ) is a complete Kähler metric measure space, or even gradient shrinking Kähler-Ricci soliton. Let v(z 2 , · · · , v n ) be polynomials in z 2 , · · · , z n , and put u = vz 1 . Then we see ∆ F u = −u. 6