The Log-Sobolev inequality for a submanifold in manifolds with asymptotic non-negative intermediate Ricci curvature

We prove a sharp Log-Sobolev inequality for submanifolds of a complete non-compact Riemannian manifold with asymptotic non-negative intermediate Ricci curvature and Euclidean volume growth. Our work extends a result of Dong-Lin-Lu which already generalizes Yi-Zheng: arXiv:2104.05045 and Brendle: arXiv:1908.10360v3.


Introduction
Log-Sobolev inequalities have been already studied in mathematical physics and information theory in the 50's ( [29], [14]) then it has been extensively studied in analysis and geometry (see e.g.[17], [16], [24], [22], [13], [10], [25], [26]), with recent developments including their applications to heat kernel and Hamilton-Jacobi equations, inequalities in convex geometry, and their role in optimal transport.Brendle [5] obtained a sharp log-Sobolev inequality for submanifolds in Euclidean space, improving [13], using the Alexandroff-Bakelman-Pucci (ABP) method and optimal transport.Later Yi and Zheng [34] extended this result to Riemannian manifolds with non-negative sectional curvature and Euclidean volume growth conditions.Subsequently Dong-Lin-Lu [12] further extended this to non-negative asymptotic sectional curvature.In this paper, we prove that the logarithmic Sobolev inequality holds under non-negative intermediate Ricci curvature.Furthermore, we extend our result to the asymptotic non-negative intermediate Ricci curvature by introducing error terms that depends only on the curvature's decay at infinity, as in [12].
Let (M n , g) be a complete non-compact Riemannian manifold.For 1 ≤ k ≤ n − 1 we consider a k-dimensional subspace P of a tangent space T x M at x ∈ M. Given any tangent vector v ∈ T x M with v⊥P , the k-intermediate Ricci curvature (kth-Ricci curvature) with respect to P in the direction of v is defined as Jihye Lee: UC Santa Barbara, Department of Mathematics, email: jihye@ucsb.edu.Fabio Ricci: UC Santa Barbara, Department of Mathematics, email: FabioRicci@ucsb.edu.
where {e 1 , . . ., e k } is an orthonormal basis of P .The k-intermediate Ricci curvature interpolates between Ricci and sectional curvature.A Riemannian manifold has nonnegative k-Ricci curvature if Ric k (P, v) ≥ 0 for any x ∈ M, k-dimensional subspace P ⊂ T x M, and a unit tangent vector v ∈ T x M perpendicular to P (note that some papers [30,21] require the stronger condition that this holds for all v, not necessary just the perpendicular ones).This condition is denoted by Ric k ≥ 0. In particular, it exhibits a monotonicity property: if n ≤ m, then Ric n ≥ 0 implies Ric m ≥ 0. This curvature condition has been well studied to explore the gap of the global results with sectional curvature bounds and Ricci curvature bounds, see for example: [28,32,9,19,18,27,21,23,30].Our work follows this spirit and generalizes Yi-Zheng's result with sectional curvature lower bound (Theorem 1.1, [34]) and the corresponding asymptotic extension by Dong-Lin-Lu (Theorem 1.1, [12]).Ketterer-Mondino [21] noted that it is possible to characterize lower bounds of k-intermediate Ricci curvature via optimal transport.We will adopt this approach, developed in [23] and [30], to prove our main result.
Let M be an n-dimensional non-compact Riemmanian manifold with non-negative Ricci curvature.The asymptotic volume ratio of M is defined as where p is some fixed point in M, ω n is the volume of the unit ball in Euclidean space R n , and |B r (p)| is the volume of a ball of radius r centered at p in M. The Bishop-Gromov volume comparison assures that the limit exists, it does not depend on the choice of p and that θ ≤ 1.We say that M has Euclidean volume growth if θ > 0.
In this paper we are using the definition of asymptotically non negative sectional ( resp.Ricci) curvature given in Abresch [1] (see also [2], [20], [35]).Zhu (see theorem 2.1 in [36]) proved the equivalent of the classical Bishop-Gromov volume comparison Theorem with Ric ≥ 0 by replacing R n with a different model space.We will need to define the equivalent of the usual asymptotic volume ratio.Definition 1.1 (Abresch [1]).An n-dimensional complete non-compact Riemannian manifold (M, g) with base point o has asymptotically non-negative Sectional curvature (Ricci curvature, respectively) if and only if there exists any non-negative, nonincreasing function λ : [0, ∞) → [0, ∞) such that the following holds: The first condition also implies that b 1 (λ) = ´∞ 0 λ(t)dt < ∞.To extend this notion of asymptotically non-negative sectional curvature to intermediate k-Ricci curvature, one can simply replace the second condition with: More precisely, Ric k (P, v) ≥ −kλ(d(o, p)) for any k dimensional subspace P ⊂ T p M and unit tangent vector v perpendicular to P .
The usual non-negative sectional curvature (Ricci or Ric k respectively) condition is equivalent to requiring λ ≡ 0. It's immediately evident from the definition to see that this class of manifolds includes those with non-negative sectional curvature (Ricci or Ric k respectively) outside a compact set and asymptotically flat manifolds as well.Furthermore, the monotonicity of the intermediate curvature still holds: if In this setting, we also replace θ, the asymptotic volume ratio of M, with another similar quantity that will keep track of the geometry at infinity.To achieve this, we define h(t) as the unique solution of the following ODE: We now define the asymptotic volume ratio of M with respect to h by In the above definition, |B r (o)| represents the volume of the ball of radius r centered at o in M and ω n represents the volume of the unit ball in R n .In [36], to prove the corresponding version of the Bishop-Gromov volume comparison in this setting, Zhu noted that the following function of r is non-increasing: This ensures that the limit exists and θ h is well-defined.Similarly, we say that M has Euclidean volume growth if θ h > 0. In particular, when a manifold has non-negative intermediate Ricci curvature, λ(t) ≡ 0, h(t) = t, and θ h is the usual asymptotic volume ratio θ.In the second section of this paper we will show a connection between θ h and an integral of a certain Guassian function.To state our theorem, we first define, for any non-negative t, a decreasing function P (t): where θ h is the asymptotic volume ratio of M with respect to h and H is the mean curvature vector of Σ and α = (n + m − 1)(2r Similar to the Michael-Simon-Sobolev inequality, the inequality (1) contains a term that involves the mean curvature and depends on the asymptotic volume ratio of M. If λ ≡ 0, the usual non-negative Ric k condition, in other words b 0 = b 1 = 0, then our Theorem 1.2 becomes: Corollary 1.3.Let M n+m be a complete, non-compact manifold of dimension n + m with Euclidean volume growth and Ric k ≥ 0, where k = min{n − 1, m − 1}.Let Σ n be a n-dimensional compact submanifold without a boundary.Then, for any positive smooth function f , we have where θ is the asymptotic volume ratio of M and H is the Mean curvature vector of Σ.
This gives the following two consequences by choosing f ≡ 1 in (2) and apply the argument in Section 4 of [34].In the case of any hypersurface (m = 1) under non-negative Ricci curvature condition, these two corollaries have been proved by Agostiniani-Fogagnolo-Mazzieri (see Theorem 1.6 in [3]).We do not recover this case in our result since k = min{n−1, m−1} would be zero.On the other hand, for higher dimension and codimension, these corollaries are also new in the usual non-negative k-intermediate Ricci curvature setting.When k = 1, the standard Ric 1 ≥ 0 is equivalent to the non-negative sectional curvature condition.Thus, we recover the corollaries in Yi-Zheng (Corollaries 1.1 and 1.2 in [34]).
The same argument (Section 4, [34]) cannot be applied in the case of asymptotic non-negative k-intermediate Ricci curvature setting because the error term we have introduced is not scale-invariant and will blow up if we blow down the metric as in [34].
Note that the previous two corollaries do not hold for Ric ≥ 0 total dimension 4; a counterexample is the Eguchi-Hanson metric on T S 2 , which is Ricci flat with Euclidean volume growth and has a totally geodesic submanifold S 2 .See, for example, [4,Page 270].
The proof of Theorem 1.2 relies on the Alexandroff-Bakelman-Pucci (ABP) method, a standard technique in Euclidean space [15].Cabrè [7] extended this method to manifolds.The classical estimate in R n uses affine functions, but on manifolds, hyperplanes are replaced with paraboloids interpreted as graphs of the distance squared relative to a point.If a lower bound condition on the curvature is added, this method has been successfully used to prove geometric inequalities (see for example, [33,31]).In particular, Brendle [6] used this method to prove a monotonicity result in terms of the Jacobian of a certain transport map, while Wang [30] and Ma-Wu [23] independently generalized this method to k-intermediate Ricci curvature.It will be a key ingredient in our proof.
Acknowledgements: The authors gratefully acknowledge the valuable insights and guidance provided by their advisor Guofang Wei through helpful discussions.We would like to thank Frank Morgan for the references [29] and [14].We would also like to acknowledge Lingen Lu for bringing the manuscript [12] to our attention.Their contribution allowed us to simplify Section 2 in the first version of our work.And thanks Jing Wu for reaching us with the updated version of their paper [23] and useful comments.

Asymptotic Estimate
In this section we recall an estimate for θ h , which will be used in the final steps of the proof in the next section.We first recall: Lemma 2.1 (Lemma 2.1, [34]).Let M be a complete non-compact Riemannian manifold of dimension N with non-negative Ricci curvature.Then, for any p ∈ M.
The proof is based on the Bishop-Gromov volume comparison.However, in our asymptotic case, we utilize the inequality r ≤ h(r) instead.The upper bound for h, as provided in [36], is h(r) ≤ e b 0 r.Generally, the presence of b 0 = 0 precludes us from replacing r N with h N in the above lemma.Following Dong-Lin-Lu [12], it is preferable to replace r N with rh N −1 , doing so, we first recall the definition of P (t): Lemma 2.2 (Lemma 2.2, [12]).Let M be a complete non-compact Riemannian manifold of dimension N with asymptotic non-negative Ricci curvature.Then where o is the base point of M.
This was proved in [12], we are giving the same proof of this statement also here for completeness: Proof.First notice that we can apply De L'Hopital's rule to the limit of θ h and obtain where we used the monotone convergence theorem when we pass the limit under the integral sign.
For the next lemma, we aim to substitute the base point o on the right-hand side of inequality (3) with ψ(x), where ψ is a Borel map whose image is compact.where ψ : M → K is any Borel map where K ⊂ M a compact subset.
The above lemma is proven, similar to the proof of Lemma 2.2 in [34], by utilizing the triangle inequality.

Proof of Theorem 1.2
In this section, we prove log Sobolev inequality with non-negative asymptotic intermediate Ricci curvature by following the papers [11,12,34].
Let M n+m be a complete non-compact manifold of dimension n + m with asymptotically non-negative Ric k and Euclidean volume growth, where k = min{n − 1, m − 1}.Assume that Σ n is an n-dimensional compact submanifold without a boundary and let f be any positive smooth function.
We first assume that Σ is connected, which is needed for the existence of a solution to a differential equation.The inequality ( 1) is invariant when we scale a function f .Thus, by scaling, we may assume that Indeed, we can scale the function f to satisfy the equation ( 4) because log(cf ) can be any real number by choosing a suitable constant c.The left-hand-side of equation ( 4) comes from the inequality (1).Thus, it suffices to show that In order to prove Isoperimetric or Sobolev inequalities using the ABP method (see [8] and related literature), we need to consider a suitable PDE on a submanifold Σ.To apply our assumption (4), we consider a differential equation as follows: where ∇ Σ is the induced Levi-Civita connection on Σ.We do not need t o specify a boundary condition since ∂Σ = ∅.Using standard PDE theory (see [15] Chapter 6), since f is a positive function, the operator u → div Σ (f ∇ Σ u) has Fredholm index 0. Hence, we can find a smooth solution u : Σ → R. We define a contact set A r as the set of all points (x, ȳ) ∈ T ⊥ Σ satisfying for all x ∈ Σ as introduced in [6].This is a natural extension of the Cabrè's idea [7] to the submanifold case.Let Φ t : T ⊥ Σ → M be a map defined by Φ t (x, y) = exp x (t∇ Σ u(x) + ty).
Then we have the following lemma, for which we will recall the proof for the reader's convenience.
As we see in the above proof, we can define a map ψ : M → Σ as ψ(p) is the point where the function ru(x) + 1 2 d(x, p) 2 attains its minimum.Then by Lemma 3.1, we have where H 0 is the counting measure.The left hand side of the inequality ( 7) is related to the asymptotic volume ratio θ h as we showed in Lemma 2.2.Also, we apply Area formula with the map Φ r : A r → M to the right hand side of (7).That is, ˆM ˆ{(x,y)∈Ar|Φr(x,y)=p} e −( d(x,Φr (x,y)) 2r We now want to estimate the integrand in the right hand side of (8).Take any r > 0 and (x, ȳ) in A r .Define a geodesic γ(t) = exp x(t∇ Σ u(x) + tȳ) on [0, r].Let {e 1 , . . ., e n } be an orthonormal basis of T xΣ such that Hess Σ u(e i , e j ) − II(e i , e j ), ȳ is diangonal.Let us denote E i by a parallel transport vector field of e i along γ for each 1 ≤ i ≤ n.Take any orthonormal frame {e n+1 , . . ., e n+m } of T ⊥ Σ near x such that ∇ e i e α , e β = 0 at x for all 1 ≤ i ≤ n and n + 1 ≤ α, β ≤ n + m.
For each 1 ≤ i ≤ n, let us consider a Jacobi field X i (t) along γ(t) with the initial conditions i (0), e j = Hess Σ u(e i , e j ) − II(e i , e j ), ȳ for all 1 ≤ j ≤ n X ′ i (0), e α = II(e i , ∇ Σ u(x)), e α for all n + 1 ≤ α ≤ n + m.
For each n + 1 ≤ α ≤ n + m, consider a Jacobi field X α (t) along γ(t) with the initial conditions X α (0) = 0 and X ′ α (0) = e α .Define (n + m) × (n + m) matrices P (t) and S(t where tr n S(t) and tr m S(t) are defined as a partial trace of S(t).Let us note that However, we cannot apply curvature assumption directly because γ′ (t) may not be perpendicular to the plane spanned by E 1 (t), . . ., E n (t).Since E i (t) is a parallel vector field along γ(t), it is enough to consider the angle at t = 0, denoted by a, between the vector γ′ (0) and the tangent plane T xΣ.Since γ′ (0) = ∇ Σ u(x) + ȳ, depending on the vectors ∇ Σ u(x) and ȳ the angle is determined.
Lemma 3.2.For (x, y) ∈ A r , we have: Since f (x) = 0 for all x ∈ Σ, we have Note that and Combining identities ( 12), (13), and ( 14), we obtain Lemma 3.3.For (x, ȳ) ∈ A r \ {(0, 0)}, we have: , Proof.Combining inequality (11) and Lemma 3.2, we obtain det P (t) By the inequality λ ≤ e λ−1 , we have  The above simple inequality will take care of the right hand side when we apply inequality (1) on each connected component individually.And this completes our proof of Theorem 1.2.

Corollary 1 . 4 .
Let M n+m be a complete, non-compact manifold of dimension n + m with non-negative intermediate k-Ricci curvature, where k = min{n−1, m−1}.Suppose M has a Euclidean volume growth.Then, there is no closed minimal submanifold of dimension n in M. Corollary 1.5.Let M n+m be a complete, non-compact manifold of dimension n + m with non-negative intermediate k-Ricci curvature, where k = min{n−1, m−1}.Suppose there is a closed minimal submanifold of dimension n in M.Then, the asymptotic volume ratio of M is zero.