On the spectrum of the weighted p-Laplacian under the Ricci-harmonic flow

This paper studies the behaviour of the spectrum of the weighted p-Laplacian on a complete Riemannian manifold evolving by the Ricci-harmonic flow. Precisely, the first eigenvalue diverges in a finite time along this flow. It is further shown that the same divergence result holds on gradient shrinking and steady almost Ricci-harmonic solitons under the condition that the soliton function is nonnegative and superharmonic. We also continue the program in (Abolarinwa, Adebimpe and Bakare in J. Ineq. Appl. 2019:10, 2019) to the case of volume-preserving Ricci-harmonic flow.


Introduction
In this paper we aim at studying the properties of the spectrum of the weighted p-Laplacian on a complete Riemannian manifold with evolving geometry. It is a well known feature that spectrum as an invariant quantity evolves as the domain does under any geometric flow. Throughout, we will consider an n-dimensional complete Riemannian manifold (M, g, dμ) equipped with weighted measure dμ = e -φ dv and potential function φ ∈ C ∞ (M, dμ), whose metric g = g(t) evolves along either the Ricci-harmonic flow or volume-preserving Ricci-harmonic flow. Firstly, we extend results in [8] to the case of volume-preserving Ricci-harmonic flow. We will obtain a variation formula for the first eigenvalue and show that it is monotonically increasing under this setup. Secondly, we study maximal time behaviour of the first eigenvalue. It is found that the bottom of the spectrum diverges in a finite time of the flow existence. We observe the same result for the behaviour of the evolving spectrum on a class of self-similar solutions, called gradient almost Ricci-harmonic solitons.
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The Ricci-harmonic flow
The pair (g = g(t), φ = φ(t)) is said to be a Ricci-harmonic flow if it satisfies the system of quasilinear parabolic equations subject to the initial condition (g(0), φ(0)) = (g 0 , φ 0 ). Here φ : M × [0, ∞) → R is a oneparameter family of smooth functions, at least C 2 in x and C 1 in t, ⊗ is the tensor product, Rc is the Ricci curvature tensor of (M, g), ∇ is the gradient operator, α is a nonincreasing constant function of time, bounded below by α n > 0 in time, and is the Laplace-Beltrami operator on M. The system (1.1) was first studied by List [20] with a motivation coming from general relativity. It was generalized by Müller [21] to the situation where φ : (M, g) → (N, h) ((N, h) is a compact Riemannian manifold endowed with a static metric h) and φ satisfies the heat flow for a harmonic map [15]. System (1.1) generalizes the Ricci flow [16] for the case φ is a constant. For a detailed discussion on the Ricci flow, see [12,13].

The almost Ricci-harmonic soliton
Let σ = σ (x) : M → R be a smooth function. We call the tuple (g, φ, f , σ ) a gradient almost Ricci-harmonic soliton if it satisfies the coupled system of nonlinear elliptic equations for some smooth function f on M. Here, we assume σ ≥ 0 and the tuple (g, φ, f , σ ) is said to be shrinking when σ is positive or steady when σ is null. If ∇f is a Killing vector field or f is a constant, we say that the soliton is trivial and the underlying metric is harmonic Einstein. If the soliton function σ is constant, we have Ricci-harmonic solitons, which are special solutions to (1.1) via scaling and diffeomorphism. These solutions occur as singularity models or blow-up limits for the flow. Almost Ricci-harmonic solitons are generalization of Ricci solitons, Einstein metrics, harmonic Einstein metrics, all of which are very useful in geometry and theoretical physics. For a detailed background on Ricci solitons, see [10]; for Ricci-harmonic solitons, see [3,20,21]; and for almost Ricci harmonic solitons, see [4,6,7,9].
In recent years, obtaining information about behaviours of eigenvalues of geometric operators on evolving manifolds has become a topic of concern among geometers since this information usually turns out to be useful in the study of geometry and topology of the underlying manifolds. Perelman's preprint [22] is fundamental in this respect. Cao [11] and Li [18] extended Perelman's result with or without any curvature assumption. Recently, [8] was motivated by the first author's papers [5] and [1] where he studied the evolution and monotonicity of the first eigenvalue of the p-Laplacian and weighted Laplacian, respectively. In [14], Di Cerbo proved that the first eigenvalue of Laplace-Beltrami operator on a 3-dimensional closed manifold with positive Ricci curvature diverges as t → T under the Ricci flow. The authors in [26] obtained a similar result under 3-dimensional Ricci-Bourguignon flow. In [5] the first author proved the same result for the weighted Laplacian under the Ricci-harmonic flow. Motivated by [14] and [5], we will show the same result for the eigenvalue of the weighted p-Laplacian under the Ricci-harmonic flow and on gradient almost Ricci-harmonic soliton for p = 2. Meanwhile, we will first extend the results of [8] to the case of a volume-preserving flow.

Preliminaries
Throughout this paper, (M, g) will be taken to be a closed Riemannian manifold. The Riemannian metric g(x) at any point x ∈ M is a bilinear symmetric positive definite matrix. As in [8], we denote a symmetric 2-tensor by Sc := Rc -α∇φ ⊗ ∇φ and its metric trace by S := R -α|∇φ| 2 , where R is the scalar curvature of (M, g) and We denote the Laplace-Beltrami operator on (M, g) by . We denote dv as the Riemannian volume measure on (M, g) and dμ := e -φ(x) dv, the weighted volume measure, where φ ∈ C ∞ (M).

The weighted p-Laplacian
Let f : M → R be a smooth function, for p ∈ [1, +∞). The weighted p-Laplacian on smooth functions is defined by When p = 2, this is just the Witten Laplacian, and when φ is a constant, it is just the p-Laplacian. See [8] for detailed descriptions of Witten Laplacian and p-Laplacian.

The minimax principle
The minimax principle also holds for the weighted p-Laplacian where its first nonzero eigenvalue is characterized as follows:  5) or equivalently, for all ψ ∈ C ∞ 0 (M) in the sense of distributions. In other words, we say that λ is an eigenvalue of p,φ and f ∈ W 1,p is the corresponding eigenfunction if the pair (λ, f ) satisfies (1.5). Then (1.6) implies implying λ = M |∇f | p dμ since M |f | p dμ = 1. Interested readers can see the book [23] for a detailed discussion on the spectral theory.

Linearized operator
As in [8,Sect. 3], we define the linearized operator of the weighted p-Laplacian on a function h ∈ C ∞ (M) pointwise at the points ∇h = 0, which is strictly elliptic in general at these points for a smooth functionf on M, where G can be viewed as a tensor defined as G := Id + (p -2) ∇h⊗∇h |∇h| 2 . Note that the sum of the second-order parts of L φ is L φf := |∇h| p-2 φf + (p -2)|∇h| p-4 Hessf (∇h, ∇h) = p,φf .
The weighted p-Laplacian degenerates at points ∇f = 0 for p = 2. In this case the εregularization technique is usually applied by replacing the linearized operator with its approximate operator. Given ε > 0, an approximate operator L φ,ε := p,φ,ε for a smooth function f ε is defined by where A ε = |∇f ε | 2 + ε. Define the G ε norm · G ε for every smooth 2-symmetric tensor V ij by Then the G ε trace of V ij is In particular, Hence, This regularization procedure has also been used in [17,24].

Variation of λ 1 (t) under volume-preserving flow
This section uses the variation formula for λ 1 (t) to establish its monotonicity under the normalized Ricci-harmonic flow. Based on the argument in [8,Sect. 4], we shall assume that λ 1 (f (t), t) = λ 1 (t) and that f (t) and 2) on a closed manifold, define a general smooth function as follows: where f (t) is a smooth function satisfying the normalization condition for all times t ∈ [0, ∞).  2). Then for any f ∈ C ∞ (M), we have the following formulas: Proof The proof follows from [8, Lemma 3.1] by using ∂ ∂t g ij = 2S ij -2r n g ij .
The following is the main theorem of this section. . Now we apply the assumptions S ij ≥ β(S + φ)g ij and φ ≥ 0, and use the condition that S min (0) > 0 in the variation formula (2.2) to get By a similar argument as in [8], we obtain in any sufficiently small neighbourhood of t 0 . Integrating with respect to t ∈ [t 1 , t 2 ], t 1 < t 2 yields A simple calculus gives

Behaviour of λ 1 (t) at the maximal time
In [14], Di Cerbo proved that the first eigenvalue of Laplace-Beltrami operator on a 3dimensional closed manifold with positive Ricci curvature diverges as t → T under the Hamilton's Ricci flow. In [5] the author proved the same result for the weighted Laplacian under Ricci-harmonic flow. Motivated by [14] and [5], we will show the same result for the eigenvalue of the weighted p-Laplacian under the Ricci-harmonic flow. Our derivation will be via weighted p-Reilly formula.
for f ∈ C ∞ (M) and Before we state the main result of the section, we remark that it has been proved in [ where W (t) is the Weyl part of the Riemannian tensor, under the extended Ricci flow for the case n ≥ 3 and T < ∞. Also in this case, |∇φ| 2 is uniformly bounded. Observe that if one assumes (3.2), then one can easily deduce that   Since p,φ f = p f -|∇f | p-2 ∇φ, ∇f , and using an elementary inequality of the form (a + b) 2 ≥ 1 1+s a 2 -1 s b 2 for s > 0, we obtain the following inequality: