Abstract
We prove Michael-Simon type Sobolev inequalities for n-dimensional submanifolds in \((n+m)\)-dimensional Riemannian manifolds with nonnegative kth intermediate Ricci curvature by using the Alexandrov-Bakelman-Pucci method. Here \(k=\min (n-1,m-1)\). These inequalities extend Brendle’s Michael-Simon type Sobolev inequalities on Riemannian manifolds with nonnegative sectional curvature Brendle (Commun. Pure Appl. Math. 76(9), 2192–2218 (2022)) and Dong-Lin-Lu’s Michael-Simon type Sobolev inequalities on Riemannian manifolds with asymptotically nonnegative sectional curvature Dong et al. (Sobolev inequalities in manifolds with asymptotically nonnegative curvature, 2022) to the k-Ricci curvature setting. In particular, a simple application of these inequalities gives rise to some isoperimetric inequalities for minimal submanifolds in Riemannian manifolds.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The classical isoperimetric problem is to find the largest possible area for a planar domain with given perimeter. The attendant isoperimetric inequality also has a long history and has been developed in many different settings. One of the intriguing directions is to prove isoperimetric inequality for minimal surfaces (cf. [3,4,5,6,7,8]). When we turn to minimal submanifolds in \(\mathbb {R}^{n+1}\), the isoperimetric inequality is closely related to the famous Michael-Simon Sobolev inequality (cf. [9]). In a recent breakthrough, inspired by the Alexandrov-Bakelman-Pucci technique in the proof of isoperimetric inequality (cf. [10, 11]), Brendle [12] proved an elegant Michael-Simon-type inequality. When the codimension is at most 2, this solves the long-standing conjecture that the (sharp) isoperimetric inequality holds for minimal submanifolds in \(\mathbb {R}^{n+1}\). Moreover, Brendle [1] generalized the Michael-Simon type inequality as well as the isoperimetric inequality to minimal submanifolds in Riemannian manifolds with nonnegative sectional curvature. For recent progress about isoperimetric inequality for minimal submanifolds, we refer to [1, 12, 13] and references therein.
Brendle’s work [1] has been extended to several different curvature settings. For example, Johne [14] considered the case of nonnegative Bakry-Émery Ricci curvature, and Dong-Lin-Lu [2] considered the case of asymptotically nonnegative curvature. In this paper, we focus on the intermediate Ricci curvature (or simply k-Ricci curvature for the kth intermediate Ricci curvature), which can be regarded as the average of some sectional curvatures. To the best of our knowledge, the notion of k-Ricci curvature was introduced by Bishop and Crittenden (cf. [15], p.253). Since the k-Ricci curvature interpolates between the sectional curvature and the Ricci curvature, it is natural to consider k-Ricci bounds as a weaker curvature condition instead of the sectional curvature bound. Some early results involving k-Ricci bounds were obtained by Galloway [16], Wu [17] and Shen [18], etc. Recently there has been an increasing interest in the relations between k-Ricci bounds and the topology and geometry of manifolds (cf. [19,20,21,22]). Remark that the link between intermediate Ricci curvatures and optimal transport was discussed in [23, 24]. By noticing the definition of intermediate Ricci curvature, there is different from the above usual one.
One of our main results is the following theorem which extends Theorem 1.4 in [1] to the k-Ricci setting.
Theorem 1.1
Let M be a complete noncompact \((n+m)\)-dimensional Riemannian manifold with nonnegative k-Ricci curvature, where \(k=\min (n-1,m-1)\). Let \(\Sigma \) be a compact n-dimensional submanifold of M (possibly with boundary \(\partial \Sigma \)), and let f be a positive smooth function on \(\Sigma \). If \(m\ge 2\), then
where \(\theta \) denotes the asymptotic volume ratio of M and H denotes the mean curvature vector of \(\Sigma \).
By putting \(f\equiv 1\) and \(H\equiv 0\) in Theorem 1.1, we obtain an isoperimetric inequality.
Corollary 1.2
Let M be a complete noncompact \((n+m)\)-dimensional Riemannian manifold with nonnegative k-Ricci curvature, where \(k=\min (n-1,m-1)\). Let \(\Sigma \) be a compact minimal n-dimensional submanifold of M with boundary \(\partial \Sigma \). If \(m\ge 2\), then
where \(\theta \) denotes the asymptotic volume ratio of M. In particular, if \(m=2\), then
Remark 1.3
We should mention that Wang [24] recently provided an optimal transport proof of the Michael-Simon inequality using a different definition for the intermediate Ricci curvature.
Another main result is a generalization of Theorem 1.1, which extends Theorem 1.5 in [2] to the case that the ambient manifold has asymptotically nonnegative k-Ricci curvature.
Theorem 1.4
Let M be a complete noncompact \(n+m\)-dimensional Riemannian manifold of asymptotically nonnegative k-Ricci curvature with respect to a base point o in M, where \(k=\min (n-1,m-1)\). Let \(\Sigma \) be a compact n-dimensional submanifold of M (possibly with boundary \(\partial \Sigma \)), and let f be a positive smooth function on \(\Sigma \). If \(m\ge 2\), then
where \(r_0=\max \{d(o,x)|x\in \Sigma \}\), \(\theta _h\) denotes the asymptotic volume ratio of M with respect to h, H denotes the mean curvature vector of \(\Sigma \), and \(b_0\), \(b_1\) are defined by (2.1) and (2.2).
In particular, we can also obtain an isoperimetric type inequality by putting \(f\equiv 1\) and \(H\equiv 0\) in Theorem 1.4.
Corollary 1.5
Assuming same conditions as in Theorem 1.4, we have
This paper is organized as follows. Section 2 contains some basic concepts. In Sect. 3, we deal with the k-Ricci curvature and prove Theorem 1.1. Using similar argument, we prove a Michael-Simon type Sobolev inequality (Theorem 1.4) for manifolds with asymptotic nonnegative k-Ricci curvature in Sect. 4.
2 Preliminaries
2.1 k-Ricci Curvature
Let M be a Riemannian manifold. The k-Ricci curvature is the average of sectional curvature over a k-dimensional subspace of the tangent space. Let \(X\in T_p M\) be a unit tangent vector and \(V\subset T_p M\) be a k-dimensional subspace such that \(X\perp V\). The k-Ricci curvature of (X, V) is defined by
where R is the Riemann curvature tensor and \(\{e_i\}\) is an orthonormal basis of V. It is worth noting that \(Ric_1\) is just sectional curvature and \(Ric_{n-1}\) is just Ricci curvature. We say a manifold M has nonnegative k-Ricci curvature (denoted by \(Ric_k\ge 0\)) if at each point \(p\in M\), for any unit tangent vector \(X\in T_p M\) and k-dimensional subspace V such that \(X\perp V\), we have \(Ric_k(X,V)\ge 0\).
Remark 2.1
There is another definition for the intermediate Ricci curvature, say \(\widetilde{Ric}_k(X,V)\), without the restriction \(X\bot V\) (e.g., [23, 24]). We remark that \(\widetilde{Ric}_k \ge 0\) is equivalent to \(Ric_{k-1} \ge 0\) for each \(k\ge 2\).
2.2 Asymptotic k-Ricci Curvature
The notion of asymptotically nonnegative curvature was first introduced by Abresch [25]. Let \(\lambda :[0,+\infty )\rightarrow [0,+\infty )\) be a nonnegative nonincreasing continuous function satisfying
and
A complete noncompact n-dimensional Riemannian manifold (M, g) is said to have asymptotically nonnegative Ricci curvature (sectional curvature, respectively) if there is a base point \(o\in M\) such that
at each point \(q\in M\). Similarly, we can define the concept of asymptotically nonnegative k-Ricci curvature in the sense that there exists a base point \(o\in M\) such that
at each point \(q\in M\). Remark that if \(k_1\le k_2\) and \(Ric_{k_1} \ge -k_1\lambda (d(o,q))\), then \(Ric_{k_2}\ge -k_2\lambda (d(o,q))\). By definition, a manifold whose (Ricci, sectional or k-Ricci, respectively) curvature is either nonnegative outside a compact domain or asymptotically flat has asymptotically nonnegative (Ricci, sectional or k-Ricci, respectively) curvature.
2.3 Asymptotic Volume Ratio
Let M be a complete noncompact n-dimensional Riemannian manifold. The asymptotic volume ratio \(\theta \) can be regarded as the ratio of the volume of geodesic ball in M to the volume of Euclidean ball in \(\mathbb {R}^n\) with same, arbitrary large radius. Precisely, the asymptotic volume ratio \(\theta \) is defined as
where q is an arbitrary fixed point in M and \(B^n\) is the unit ball in \(\mathbb {R}^{n+1}\). If M has nonnegative Ricci curvature, the Bishop-Gromov volume comparison theorem indicates that \(\theta \) exists and \(\theta \le 1\).
Similarly, let h(t) be the unique solution of
where \(\lambda \) is the nonnegative function given in Section 2.2. The asymptotic volume ratio of M with respect to h is defined by
3 Manifolds With Nonnegative k-Ricci Curvature
In this section, we assume that (M, g) is a complete noncompact \((n+m)\)-dimensional Riemannian manifold with nonnegative k-Ricci curvature, where \(k=\min (n-1,m-1)\). We also assume that \(\Sigma \) is a compact n-dimensional submanifold of M (possibly with boundary \(\partial \Sigma \)), and f is a positive smooth function on \(\Sigma \). Let \({\bar{D}}\) and \(D^\Sigma \) denote the Levi-Civita connection on (M, g) and the induced connection on \(\Sigma \), respectively. Let \({\bar{R}}\) denote the Riemann curvature tensor on (M, g). For any tangent vector fields X, Y on \(\Sigma \) and normal vector field \(\eta \) along \(\Sigma \), the second fundamental form II of \(\Sigma \) is given by
We only need to prove Theorem 1.1 in the case that \(\Sigma \) is connected. By scaling, we assume that
Since \(\Sigma \) is connected, there exists a solution u to the following Neumann boundary problem.
where \(\nu \) is the outward conormal to \(\partial \Sigma \). By standard elliptic regularity theory, \(u\in C^{2,\gamma }(\Sigma )\) for each \(0<\gamma <1\) (cf. Theorem 6.30 in [26]).
As in [1], we define
For each \(r>0\), we denote by \(A_r\) the contact set, that is the set of all points \(({\bar{x}},{\bar{y}})\in U\) with the property that
for all x in \(\Sigma \). Moreover, for each \(r>0\) we define the transport map \(\Phi _r: T^\perp \Sigma \rightarrow M\) by
for all \(x\in \Sigma \) and \(y\in T^\perp _x \Sigma \). For each \(0<\gamma <1\), since u is of class \(C^{2,\gamma }\), \(\Phi _r\) is of class \(C^{1,\gamma }\).
The following four lemmas are due to Brendle [1]. Their proofs are independent of the curvature condition, so they also hold in our setting and we omit the proofs here.
Lemma 3.1
(Lemma 4.1 in [1]) Assume that \(x\in \Omega \) and \(y\in T_x^\perp \Sigma \) satisfy \(|D^\Sigma u(x)|^2+|y|^2\le 1\). Then \(\Delta _\Sigma u(x)-\langle H(x),y\rangle \le n f(x)^{\frac{1}{n-1}}\).
Lemma 3.2
(Lemma 4.2 in [1]) For each \(0\le \sigma <1\), the set
is contained in the set
Lemma 3.3
(Lemma 4.3 in [1]) Assume that \(({\bar{x}}, {\bar{y}})\in A_r\), and let \({\bar{\gamma }}(t):=\exp _{{\bar{x}}}(tD^\Sigma u ({\bar{x}})+t {\bar{y}})\) for all \(t\in [0,r]\). If Z is a smooth vector field along \({\bar{\gamma }}\) satisfying \(Z(0)\in T_{{\bar{x}}}\Sigma \) and \(Z(r)=0\), then
Lemma 3.4
(Lemma 4.5 in [1]) Assume that \(({\bar{x}}, {\bar{y}})\in A_r\), and let \({\bar{\gamma }}(t):=\exp _{{\bar{x}}}(tD^\Sigma u ({\bar{x}})+t {\bar{y}})\) for all \(t\in [0,r]\). Moreover, let \(\{e_1,\dots ,e_n\}\) be an orthonormal basis of \(T_{{\bar{x}}}\Sigma \). Suppose that W is a Jacobi field along \(\bar{\gamma }\) satisfying \(W(0)\in T_{{\bar{x}}}\Sigma \) and \(\langle {\bar{D}}_t W(0), e_j\rangle =(D^2_\Sigma u)(W(0),e_j)-\langle II(W(0),e_j),{\bar{y}}\rangle \) for each \(1\le j\le n\). IF \(W(\tau )=0\) for some \(\tau \in (0,r)\), then W vanishes identically.
Now with the preparation above, we are in a position to prove Theorem 1.1.
Proof of Theorem 1.1
Throughout the proof, we use the following notions of indices
For any \(r>0\) and \(({\bar{x}}, {\bar{y}})\in A_r\), let \(\{e_i\}_{1\le i\le n}\) be any given orthonormal basis in \(T_{{\bar{x}}}\Sigma \). We can choose geodesic normal coordinates \((x_1,\dots ,x_n)\) on \(\Sigma \) around \({\bar{x}}\) such that \(\frac{\partial }{\partial x_i}=e_i\) at \({\bar{x}}\) for each \(1\le i\le n\). Let \(\{e_\alpha \}_{n+1\le \alpha \le n+m}\) be a local orthonormal frame of \(T^\perp \Sigma \) around \({\bar{x}}\) such that \(\langle {\bar{D}}_{e_i}e_\alpha , e_\beta \rangle =0\) at \({\bar{x}}\). Now a normal vector y can be written as \(y=\sum _\alpha y_\alpha e_\alpha \) and in this sense \((x_1,\dots ,x_n,y_{n+1},\dots ,y_{n+m})\) forms a local coordinate system on the total space of the normal bundle \(T^\perp \Sigma \).
Let \({\bar{\gamma }}(t):=\exp _{{\bar{x}}}(tD^\Sigma u ({\bar{x}})+t {\bar{y}})\) for all \(t\in [0,r]\). For each \(1\le A\le n+m\), let \(E_A(t)\) be the parallel transport of \(e_A\) along \({\bar{\gamma }}\). For each \(1\le i \le n\), let \(X_i(t)\) be the unique Jacobi field along \({\bar{\gamma }}\) satisfying
For each \(n+1\le \alpha \le n+m\), let \(X_\alpha (t)\) be the unique Jacobi field along \({\bar{\gamma }}\) satisfying
Lemma 3.4 tells us that \(\{X_A(t)\}_{1\le A\le n+m}\) are linearly independent for each \(t\in (0,r)\).
Then we define two \((n+m)\times (n+m)\)-matrix \(P(t)=(P_{AB}(t))\) and \(S(t)=(S_{AB}(t))\) by
Based on the observation that
for each \(1\le A\le n+m\), we conclude that
for all \(t\in (0,r)\). Therefore, we only need to estimate \(\det P(t)\).
By the definition of the Jacobi fields \(X_A(t)\), the Jacobi equation reads
with the initial conditions
and
Moreover, since
is symmetric for each \(t\in (0,r)\), \(P'(t)P(t)^T\) is also symmetric for each t. Let Q(t) be a matrix defined by
which is symmetric for each \(t\in (0,r)\). Then the Riccati equation reads
From the asymptotic expansion of P(t)
a direct computation gives rise to
as \(t\rightarrow 0\).
By taking partial trace of Q(t), we reduce the Riccati equation (3.4) to the following two equations:
where we use the Cauchy-Schwarz inequality.
By assumption, M has nonnegative k-Ricci curvature. We claim that \(\sum _i S_{ii}(t)\) and \(\sum _\alpha S_{\alpha \alpha }(t)\) are nonnegative for any \(t\in [0,r)\).
In fact, for fixed \(t\in [0,r)\), without loss of generality, we can choose \(e_1=\frac{D^{\Sigma }u({\bar{x}})}{|D^{\Sigma }u({\bar{x}})|}\) and \(e_{n+1}=\frac{{\bar{y}}}{|{\bar{y}}|}\). Denote \(a:=\sqrt{|D^{\Sigma }u({\bar{x}})|^2+|{\bar{y}}|^2}\), then there exist an angle s and a vector field \(\xi \) along \(\bar{\gamma }\), such that
as well as
Now
Since \(\{\frac{1}{a}{\bar{\gamma }}'(t), E_2(t), \dots , E_n(t), \frac{1}{a}\xi (t), E_{n+2}(t), \dots , E_{n+m}(t)\}\) form an orthogonal basis, we can compute that
Analogously we also have \(\sum _\alpha S_{\alpha \alpha }(t)\ge 0\).
Therefore, the equations (3.6) become
A standard ODE comparison gives
Integrating (3.12) over \([\epsilon ,t]\) for \(0<\epsilon <t\) and letting \(\epsilon \rightarrow 0^+\), we obtain
for all \(t\in (0,r)\).
By Lemma 3.1, we have
As in the proof of [1], together with the above estimate, Lemma 3.2 tells us
for all \(r>0\) and all \(0\le \sigma <1\). Now dividing both sides by \(r^{n+m}\) and letting \(r\rightarrow \infty \), we have
for all \(0\le \sigma <1\). Finally, dividing both sides by \(1-\sigma \) and letting \(\sigma \rightarrow 1\), we have
Consequently, it is apparent from the scaling assumption (3.1) that
\(\square \)
4 Manifolds With Asymptotically Nonnegative k-Ricci Curvature
Proof of Theorem 1.4
By scaling, we assume that
We use the same notions as in the proof of Theorem 1.1 except that u is a solution of the following Neumann boundary problem:
Since we only change the definition of u, Lemma 3.2, Lemma 3.3, and Lemma 3.4 still hold. We also need another version of Lemma 3.1.
Lemma 4.1
(Lemma 3.1 in [2]) Assume that \(x\in \Omega \) and \(y\in T_x^\perp \Sigma \) satisfy \(|D^\Sigma u(x)|^2+|y|^2\le 1\). Then \(\Delta _\Sigma u(x)-\langle H(x),y\rangle \le n f(x)^{\frac{1}{n-1}}-2n b_1\).
By assumption, M has asymptotically nonnegative k-Ricci curvature with respect to a base point o in M. Analogous to the computation (3.10), we have
and
where s is the angle between \({\bar{\gamma }}'(t)\) and \(E_1(t)\).
Let \(\phi \) and \({\widetilde{\phi }}\) be defined by
respectively. Then with the initial condition (3.5), (3.6) reduce to
and
Next, in order to estimate \(\phi \) and \({\widetilde{\phi }}\), we should compare them with standard ODE solutions. Let \(\psi _1\) and \(\psi _2\) be solutions to the following ODEs
respectively. Then the function \(\psi :=\psi _1(\frac{1}{n}(\Delta _\Sigma u({\bar{x}})-\langle H({\bar{x}}),{\bar{y}}\rangle ))+\psi _2\) is the solution to
By a standard ODE comparison result (cf. [27], Lemma 2.4A), we have
Similarly, let \({\widetilde{\psi }}\) be the solution to
Then the standard ODE comparison (cf. [27], Lemma 2.4A) gives
Now taking (4.9), (4.11) into consideration, by definition of \(\phi \) and \({\widetilde{\phi }}\), we have
for all \(t\in (0,r)\). Integrating (4.12) over \([\epsilon ,t]\) for \(0<\epsilon <t\) and letting \(\epsilon \rightarrow 0^+\), we obtain
for all \(t\in (0,r)\). Moreover, a standard ODE comparison of (4.6), (4.7) and (4.10) gives
Furthermore, we have
and
By substituting (4.14), (4.15), and (4.16) into (4.13), together with Lemma 4.1, we have
Analogous to the proof of Theorem 1.1, Lemma 3.2 tells us
Now dividing both sides by \((n+m)\int _0^r h(t)^{n+m-1}\text {d}t\) and letting \(r\rightarrow \infty \), we have
Here we use the fact that \(h(t)\ge t\) and \({\underline{\lim }}_{t\rightarrow \infty } h'(t)\ge 1+b_0\) in the last line. Then dividing both sides by \(1-\sigma \) and letting \(\sigma \rightarrow 1\), we have
Finally, it follows from the scaling assumption 4.1 that
\(\square \)
Data availability
The data analysed during the current study are referenced in the text and publicly available.
References
Brendle, S.: Sobolev inequalities in manifolds with nonnegative curvature. Commun. Pure Appl. Math. 76(9), 2192–2218 (2022)
Dong, Y., Lin, H., Lu, L.: Sobolev inequalities in manifolds with asymptotically nonnegative curvature (2022)
Carleman, T.: Zur Theorie der Minimalflächen. Math. Z. 9(1–2), 154–160 (1921)
Reid, W.T.: The isoperimetric inequality and associated boundary problems. J. Math. Mech. 8, 897–905 (1959)
Osserman, R., Schiffer, M.: Doubly-connected minimal surfaces. Arch. Rational Mech. Anal. 58(4), 285–307 (1975)
Feinberg, J.M.: The isoperimetric inequality for doubly-connected minimal surfaces in \({ R}^{n}\). J. Anal. Math. 32, 249–278 (1977)
Chavel, I.: On A. Hurwitz’ method in isoperimetric inequalities. Proc. Amer. Math. Soc. 71(2), 275–279 (1978)
Li, P., Schoen, R., Yau, S.-T.: On the isoperimetric inequality for minimal surfaces. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11(2), 237–244 (1984)
Michael, J.H., Simon, L.M.: Sobolev and mean-value inequalities on generalized submanifolds of \(R^{n}\). Comm. Pure Appl. Math. 26, 361–379 (1973)
Cabré, X.: Elliptic PDE’s in probability and geometry: symmetry and regularity of solutions. Discrete Contin. Dyn. Syst. 20(3), 425–457 (2008)
Trudinger, N.S.: Isoperimetric inequalities for quermassintegrals. Ann. Inst. H. Poincaré C Anal. Non Linéaire 11(4), 411–425 (1994)
Brendle, S.: The isoperimetric inequality for a minimal submanifold in Euclidean space. J. Amer. Math. Soc. 34(2), 595–603 (2021)
Choe, J.: Isoperimetric inequalities of minimal submanifolds. In: Global theory of minimal surfaces, volume 2, pages 325–369. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA (2005)
Johne, F.: Sobolev inequalities on manifolds with nonnegative bakry-émery ricci curvature (2021)
Bishop, R.L., Crittenden, R.J.: Geometry of manifolds. Pure and Applied Mathematics, vol. XV. Academic Press, New York-London (1964)
Galloway, G.J.: Some results on the occurrence of compact minimal submanifolds. Manuscripta Math. 35(1–2), 209–219 (1981)
Wu, H.-H.: Manifolds of partially positive curvature. Indiana Univ. Math. J. 36(3), 525–548 (1987)
Shen, Z.M.: A sphere theorem for manifolds of positive Ricci curvature. Indiana Univ. Math. J. 38(1), 229–233 (1989)
Gu, J.-R., Xu, H.-W.: The sphere theorems for manifolds with positive scalar curvature. J. Differ. Geom. 92(3), 507–545 (2012)
Chahine, Y.K.: Volume estimates for tubes around submanifolds using integral curvature bounds. J. Geom. Anal. 30(4), 4071–4091 (2020)
Mouillé, L.: Torus actions on manifolds with positive intermediate Ricci curvature. J. Lond. Math. Soc. (2) 106(4), 3792–3821 (2022)
Reiser, P., Wraith, D.J.: Positive intermediate ricci curvature on fibre bundles (2022)
Ketterer, C., Mondino, A.: Sectional and intermediate Ricci curvature lower bounds via optimal transport. Adv. Math. 329, 781–818 (2018)
Wang, K.-H.: Optimal transport approach to michael-simon-sobolev inequalities in manifolds with intermediate ricci curvature lower bounds (2022)
Abresch, U.: Lower curvature bounds, Toponogov’s theorem, and bounded topology. Ann. Sci. École Norm. Sup. (4) 18(4), 651–670 (1985)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition
Cao, J., Wang, Y.: A Concise Course of Morden Riemannian Geometry (in chinese). China Science Publishing and Media, Beijing (2006)
Acknowledgements
The authors were partially supported by the National Natural Science Foundation of China under grants No. 11831005 and Nos. 12061131014. They would like to thank Professor Chao Qian for his helpful suggestions. They also thank Kai-Hsiang Wang for bringing their attention to [23, 24] and for helpful comments on the earlier version of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ma, H., Wu, J. Sobolev Inequalities in Manifolds With Nonnegative Intermediate Ricci Curvature. J Geom Anal 34, 93 (2024). https://doi.org/10.1007/s12220-023-01486-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-023-01486-5