Preserving Positive Intermediate Curvature

Abstract

Consider a compact manifold N (with or without boundary) of dimension n. Positive m-intermediate curvature interpolates between positive Ricci curvature ($m = 1$) and positive scalar curvature ($m = n-1$), and it is obstructed on partial tori $N^n = M^{n-m} \times \mathbb {T}^m$. Given Riemannian metrics $g, {\bar{g}}$ on $(N, \partial N)$ with positive m-intermediate curvature and m-positive difference $h_g - h_{{\bar{g}}}$ of second fundamental forms we show that there exists a smooth family of Riemannian metrics with positive m-intermediate curvature interpolating between g and ${\bar{g}}$. Moreover, we apply this result to prove a non-existence result for partial torical bands with positive m-intermediate curvature and strictly m-convex boundaries.

1 Introduction

The existence of Riemannian metrics with positive curvature implies obstructions on the topology of closed manifolds: Manifolds with topology $N^n = M^{n-1} \times \mathbb {S}^1$ do not admit metrics of positive Ricci curvature by the Theorem of Bonnet–Myers, while manifolds with topology $N^n = \mathbb {T}^n$ do not admit metrics of positive scalar curvature by the resolution of the Geroch conjecture due to Schoen and Yau [1] (for $3 \le n \le 7$ by using minimal hypersurfaces) and Gromov and Lawson [2] (by using spinors and the Atiyah–Singer index theorem).

The above results yield obstructions for positive curvature on the the partial tori $N^n = M^{n-m} \times \mathbb {T}^m$ for the limit cases $m = 1$ and $m = n-1$. Recently, Brendle et al. introduced the notion of positive intermediate curvature (see Sect. 2 for a precise definition), which interpolates between positive Ricci curvature $(m=1)$ and positive scalar curvature $(m = n-1)$. They obtained the following obstruction result on partial tori:

Theorem 1.1

(Generalized Geroch conjecture, Theorem 1.5 in [3]) Assume $n \le 7$ and $1 \le m \le n-1$. Let $N^n$ be a closed and orientable manifold of dimension n, and suppose that there exists a closed and orientable manifold $M^{n-m}$ and a map $F: N^n \rightarrow M^{n-m} \times \mathbb {T}^m$ with non-zero degree. Then the manifold N does not admit a metric with positive m-intermediate curvature.

The associated rigidity question was studied by Chu et al. [4] in ambient dimension at most five. Chen [5] extended the obstruction result to manifolds with arbitrary ends. Moreover, Xu [6] showed sharpness of the result in [3] by constructing counterexamples for dimensions $n > 7$ and $3 \le m \le n-3$. The above result was recently used by Labbi [7] to compute the Riemann invariant of products of spheres and tori.

In this work, we study the interaction of the internal geometry and the boundary geometry for metrics of positive intermediate curvature. The corresponding question for positive scalar curvature and mean curvature on the boundary dates back to work by Gromov and Lawson [8]. A similar interaction appears in the proof of the positive mass theorem by Schoen and Yau [9]—planes with positive mean curvature act as barriers for minimal hypersurfaces in the bulk region with positive scalar curvature. Related is work by Shi and Tam [10], where they proved an estimate for the integral of the mean curvature over the boundary in manifolds with non-negative scalar curvature. Miao [11] proved a positive mass theorem on manifolds with corners by performing a suitable intrinsic bending construction. A pertubation argument, which made the boundary totally geodesic while keeping the scalar curvature non-decreasing, allowed Brendle et al. [12] to construct counterexamples to the Min-Oo conjecture.

Recently, the first author proved a general result [13, Main Theorem 1] on the interaction of internal geometry and boundary geometry by suitably gluing Riemannian metrics. The result applies to a wide range of positive curvature conditions, for example to metrics with positive curvature operator, PIC 2, PIC 1 (with convex boundary), positive isotropic curvature (with two-convex boundary) and positive scalar curvature (with mean-convex boundary). For a different approach to the gluing problem, see also the thesis by Schlichting, [14].

The first result of our work extends the gluing result of the first author to m-intermediate curvature with the natural condition of m-convexity on the boundary:

Theorem 1.2

(Preserving positive m-intermediate curvature) Suppose that $N^n$ is a compact smooth manifold with smooth boundary $\partial N$ of dimension $\dim N = n$. Let $g, {\tilde{g}}$ be Riemannian metrics on N, such that $g = {\tilde{g}}$ on the boundary $\partial N$.

Then, there exists $\lambda _0 > 0$, a family of smooth Riemannian metrics $\{{\hat{g}}_{\lambda }\}_{\lambda > \lambda _0}$, and a neighborhood U of the boundary $\partial N$, such that the metric ${\hat{g}}_{\lambda }$ agrees with the metric g outside of U, and the metric ${\hat{g}}_{\lambda }$ agrees with the metric ${\tilde{g}}$ in a neighbourhood of $\partial N$. Additionally, we have ${\hat{g}}_{\lambda } \rightarrow g$ as $\lambda \rightarrow \infty$ in $C^{\alpha }$ for any $\alpha \in (0,1)$.

Moreover, let $1 \le m \le n-1$. If

• (1) The Riemannian manifolds (N, g) and $(N, {\tilde{g}})$ have positive m-intermediate curvature,

• (2) The difference $h_g - h_{{\tilde{g}}}$ is strictly m-convex (i.e. strictly m-positive),

then the Riemannian manifold $(N, {\hat{g}}_{\lambda })$ has positive m-intermediate curvature for all $\lambda > \lambda _0$.

The main ingredient in the proof is Proposition 3.1 relating the cone of positive m-intermediate curvature to the Kulkarni–Nomizu product of m-convex symmetric two-tensors.

In the second part of the paper we consider Riemannian manifolds with m-positive intermediate curvature and strictly m-convex boundary: Let us first recall the doubling lemma by Gromov and Lawson for manifolds with positive scalar curvature and strictly mean convex boundaries.

Lemma 1.3

(Doubling of positive scalar curvature metrics, Gromov and Lawson [8]) Suppose (N, g) is an orientable compact smooth Riemannian manifold with smooth boundary $\partial N$. Assume the metric g has positive scalar curvature and is strictly mean convex (i.e. $H_{\partial N} > 0$) with respect to the outward unit normal. Then the double of N carries a metric of positive scalar curvature.

This lemma (in conjunction with the nonexistence of positive scalar curvature metrics on the torus) then implies the following obstruction to positive scalar curvature on torical bands:

Theorem 1.4

(Boundaries of a torical band, Gromov and Lawson, [8]) Consider the smooth manifold with boundary $N = \mathbb {T}^{n-1}\times [-1, 1]$ and let g be a Riemannian metric on N with positive scalar curvature. Then the boundary $\partial N$ cannot be strictly mean convex.

Räde extended the above result to a scalar curvature and mean curvature comparison result on more general bands [15].

We extend the above result on scalar curvature, mean curvature and torical bands to partially torical bands by proving a generalization of the doubling lemma of Gromov and Lawson.

Theorem 1.5

(Boundaries of a partially torical band) Let $n \le 7$ and $1 \le m \le n-1$. Suppose $M^{n-m}$ is a closed orientable manifold. Consider the smooth manifold with boundary $N = M^{n-m}\times \mathbb {T}^{m-1}\times [-1, 1]$ and a Riemannian metric g with positive m-intermediate curvature on N. Then, the boundary $\partial N$ cannot be strictly m-convex.

The work is structured as follows: In Sect. 2 we introduce our notation and recall the definition of intermediate curvature. In Sect. 3 we prove an algebraic lemma connecting the cone of positive m-intermediate curvature and m-convexity. In Sects. 4 and 5 we prove the gluing result and in Sect. 6 we perform the doubling constructions.

2 Preliminaries

Let $(V, \langle \cdot , \cdot \rangle )$ be a n-dimensional real inner product space. The space of algebraic curvature tensors on V denoted by $C_B(V)$ is given by multilinear maps $R: V \times V \times V \times V \rightarrow \mathbb {R}$ with the symmetries of the curvature tensor, i.e.

$$\begin{aligned} R(v_1,v_2,v_3, v_4) = -R(v_2, v_1, v_3, v_4) \; \text {and} \; R(v_1, v_2, v_3, v_4) = R(v_3, v_4, v_1, v_2) \end{aligned}$$

for all $v_1, v_2, v_3, v_4 \in V$, and satisfying the first Bianchi identity, i.e.

$$\begin{aligned} R(v_1, v_2, v_3, v_4) + R(v_3, v_1, v_2, v_4) + R(v_2, v_3, v_1, v_4) = 0 \end{aligned}$$

for all $v_1, v_2, v_3, v_4 \in V$.

We denote the space of symmetric bilinear maps $T: V \times V \rightarrow \mathbb {R}$ by ${{\,\textrm{Sym}\,}}^2 V$. The Kulkarni–Nomizu product is defined by

for $v_1, v_2, v_3, v_4 \in V$.

Following Definition 1.1 in work of the Brendle et al. [3] we define the cone ${\mathcal {C}}_m(V)$ of non-negative m-intermediate curvature in the space of algebraic curvature tensors by

$$\begin{aligned} {\mathcal {C}}_m(V)&:= \Bigg \{ T \in C_B(V) \; \Bigg | \; \sum _{p=1}^m \sum _{q=p+1}^n T(e_p, e_q, e_p, e_q) \ge 0 \; \\&\qquad \text {for all orthonormal bases} \; \{e_i\}_{i=1}^n \; \text {of} \; V\Bigg \} \end{aligned}$$

For a Riemannian manifold $(N^n,g)$ with boundary $\partial N$ we consider its Levi-Civita connection D and its Riemann curvature tensor ${{\,\textrm{Rm}\,}}_N$ given by the formula

$$\begin{aligned} {{\,\textrm{Rm}\,}}_N(X,Y,Z,W) = -g( D_X D_Y Z - D_Y D_X Z - D_{[X,Y]} Z,W) \end{aligned}$$

for vector fields $X,Y,Z,W \in \Gamma (TN)$.

The Riemannian manifold $(N^n,g)$ has positive m-intermediate curvature, if ${{\,\textrm{Rm}\,}}_N(p) \in {{\,\textrm{Int}\,}}({\mathcal {C}}_m)$ for all $p \in N$ (compare with Definition 1.1 in [3]).

Let $\nu$ be the inward pointing unit normal vector field on the boundary $\partial N$. The scalar-valued second fundamental form $h_g: T(\partial N) \otimes T(\partial N) \rightarrow C^{\infty }(\partial N)$ of the boundary $\partial N$ with respect to the Riemannian metric g is defined by

$$\begin{aligned} h_g(X,Y) = g(\nu , D_X Y) \end{aligned}$$

for $X,Y \in \Gamma (TN)$. With this convention the scalar-valued second fundamental form is positive on the standard sphere $\mathbb {S}^n \subset \mathbb {R}^{n+1}$ with respect to the inward pointing unit normal vector $\nu = -x$.

We say that the boundary $\partial N$ is strictly m-convex (where $1 \le m \le n-1$), if the bilinear form $h_g(p)$ is m-positive for all $p \in \partial N$, i.e. if $\lambda _1 \le \lambda _2 \le \dots \lambda _{n-1}$ denote the eigenvalues of $h_g(p)$, then $\lambda _1 + \dots + \lambda _m > 0$. For $m = 1$ we recover the notion of strict convexity, and for $m = n-1$ we recover the notion of strict mean convexity.

3 Connecting m-Convexity and m-Intermediate Curvature

In this section, we prove a lemma in linear algebra, which allows us to connect the cone of positive m-intermediate curvature and strict m-convexity.

Proposition 3.1

(m-intermediate curvature cone and m-convexity) Let $(V, \langle \cdot , \cdot \rangle )$ be a n-dimensional inner product space and $S: V \times V \rightarrow \mathbb {R}$ be a symmetric bilinear form. Let $W \subset V$ be a $(n-1)$-dimensional subspace, and let $\nu \in W^{\perp }$ be a unit vector. Let $S|_W$ be the restriction of S on W. Fix $1 \le m \le n -1$. Then the bilinear form $S|_W$ is m-positive, if and only if

Proof

Suppose that the bilinear form $S|_W$ is m-positive. We extend the bilinear form $S|_W$ to a bilinear T on V by setting

$$\begin{aligned} T(v,w):= S|_W (v^{\parallel }, w^{\parallel }) \end{aligned}$$

for $v,w \in V$. Here $v^{\parallel }$ denotes the orthogonal projection from V to W.

The assumption on the m-positivity of the bilinear form $S|_W$ on W implies that the bilinear form T is $(m+1)$-positive on V.

We first want to show that T being $(m+1)$-positive implies .

Let $\{e_1, \dots , e_n\}$ be an orthonormal basis of V with respect to the inner product $\langle \cdot , \cdot \rangle$. We denote the components of the vector $\nu$ with respect to this orthonormal basis by $a_p$, i.e. $a_p = \langle \nu , e_p \rangle$.

We have

for $1 \le p, q \le n$ by definition of the Kulkarni–Nomizu product. We observe the identity

We evaluate the first term in the above sum:

We evaluate the second term in the above sum:

This implies

(1)

If $a_p = 0$ for all $m+1 \le p \le n$, then we deduce the estimate

since the bilinear form T is $(m+1)$-positive by construction and hence n-positive. Hence we deduce in this case.

Now suppose that $a_p \ne 0$ for some $m + 1 \le p \le n$. We define the unit vector

$$\begin{aligned} w:= \left( \sum _{p=m+1}^{n} a_p^2 \right) ^{-\frac{1}{2}} \sum _{q=m+1}^n a_q e_q \end{aligned}$$

With this definition we deduce from Eq. (1) the identity

The first term involving the trace ${{\,\textrm{tr}\,}}_V(T)$ is positive as above. The term in the bracket is positive, since the bilinear form T is $(m+1)$-positive, and $w \perp {{\,\textrm{span}\,}}\{e_1, \dots , e_m\}$ by construction. Hence the sum is positive and we deduce .

On the other hand, by the construction of T, the restriction of $S-T$ to the subspace W vanishes. Therefore, we may write

$$\begin{aligned} S = T + \omega \otimes \nu ^b+ \nu ^b \otimes \omega \end{aligned}$$

where $\omega$ is a suitable 1-form. Note that

by symmetry. Hence,

The other implication in the equivalence follows by taking the orthonormal basis $\{f_1, \dots , f_{n-1}, \nu \}$ of the vector space V, where $\{f_1, \dots , f_{n-1}\}$ is an orthonormal basis of the subspace W. $\square$

4 Preserving Curvature Conditions

In this section, we assume that g and ${\tilde{g}}$ are Riemannian metrics on N such that $g - {\tilde{g}} = 0$ along $\partial N$. We describe our choice of perturbation as in work of Brendle et al. [12]. We fix a neighborhood U of the boundary $\partial N$ and a smooth boundary defining function $\rho : N\rightarrow [0,\infty )$ by taking it to be the distance function from the boundary $\partial N$ with respect to the metric g. Then, we have $|D\rho |_g =1$. Since $g-{\tilde{g}}=0$ along the boundary $\partial N$, we can find a symmetric (0, 2)-tensor S such that ${\tilde{g}}=g+\rho S$ in a neighborhood of $\partial N$ and $S=0$ outside U. The scalar-valued second fundamental forms and the boundary defining function satisfy

$$\begin{aligned} \frac{1}{2}S(X,Y)= h_g(X,Y)-h_{{\tilde{g}}}(X,Y), \; \text {and} \; D^2\rho (X,Y) = -h_g(X,Y) \end{aligned}$$

for all $X,Y \in \Gamma (T (\partial N))$. This implies that the identity

$$\begin{aligned}&h_g(X,Y) - h_{{\tilde{g}}}(X,Y)=\frac{1}{2}S(X,Y)=-D^2\rho (X,Y) - h_{{\tilde{g}}}(X,Y) \end{aligned}$$

(2)

holds on the boundary $\partial N$ for all $X,Y\in \Gamma (T(\partial N))$.

We choose a smooth cut-off function $\chi :[0,\infty )\rightarrow [0,1]$ with the following properties (compare with [12, Lemma 17]):

• $\chi (s)=s-\frac{1}{2}s^2$ for $s\in [0,\frac{1}{2}]$;

• $\chi (s)$ is constant for $s\ge 1$;

• $\chi ''(s)<0$ for $s\in [0,1)$.

Moreover, we choose a smooth cut-off function $\beta :(-\infty ,0]\rightarrow [0,1]$ such that

• $\beta (s)=\frac{1}{2}$ for $s\in [-1,0]$;

• $\beta (s)=0$ for $s\in (-\infty ,-2]$.

For $\lambda > 0$ sufficiently large we define a smooth metric ${\hat{g}}_{\lambda }$ on the manifold N by the formula

$$\begin{aligned} {\hat{g}}_{\lambda }= {\left\{ \begin{array}{ll} g+\lambda ^{-1}\chi (\lambda \rho )S\quad \quad \quad \quad &{}\text {for}\quad \rho \ge e^{-\lambda ^2}\\ {\tilde{g}}-\lambda \rho ^2\beta (\lambda ^{-2}\log \rho )S\quad &{}\text {for}\quad \rho <e^{-\lambda ^2} \end{array}\right. } \end{aligned}$$

(3)

In the sequel, we will show that ${\hat{g}}_{\lambda }$ preserves positive m-intermediate curvature of g and ${\tilde{g}}$ for sufficiently large $\lambda > 0$. Note that we have the identity ${\hat{g}}_{\lambda }={\tilde{g}}$ in the region $\{\rho \le e^{-2\lambda ^2}\}$ and ${\hat{g}}_{\lambda }=g$ outside the neighbourhood U. Moreover, from the construction it follows that ${\hat{g}}_{\lambda } \rightarrow g$ as $\lambda \rightarrow \infty$ in $C^{\alpha }$ for any $\alpha \in (0,1)$.

We first derive a lower bound for the m-intermediate curvature of the metric ${\hat{g}}_{\lambda }$. We first consider the region $\{\rho \ge e^{-\lambda ^2}\}$.

Proposition 4.1

(Curvature estimates in inner gluing region) Suppose that $h_g - h_{{\tilde{g}}}$ is m-positive on the boundary $\partial N$. Let $\epsilon > 0$ be given. If $\lambda =\lambda (\epsilon , \chi )>0$ is sufficiently large, then

$$\begin{aligned} \sum _{p=1}^m\sum _{q=p+1}^n ({{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(x)(e_p, e_q, e_p, e_q) -{{\,\textrm{Rm}\,}}_{g}(x)(e_p, e_q, e_p, e_q))\ge -\epsilon \end{aligned}$$

for any ${\hat{g}}_{\lambda }$-orthonormal frame $\{e_1, \dots , e_n\}$ and any $x \in N$ in the region $\{\rho (x)\ge e^{-\lambda ^2}\}$.

Proof

We fix a point $x \in N$ such that $\rho (x)\ge e^{-\lambda ^2}$. Let $\{e_1, \dots , e_n\}$ be a geodesic normal frame around the point x with respect to the metric ${\hat{g}}_{\lambda }$. Let $\varphi$ be a two-form. We write $\varphi =\sum _{i,j}\varphi ^{ij}e_i\wedge e_j$ for coefficients $\varphi ^{ij}$, which are anti-symmetric in i and j. In the following the Einstein summation convention will be adopted freely. Since $\varphi$ is in particular a (2, 0)-tensor, $\varphi$ induces by the fundamental principle of tensor calculus a linear map $[\varphi ]: (T_x N)^*\rightarrow T_x N$ via the action $[\varphi ]w:= \varphi ^{ij}w(e_i)e_j$. Equation (5) in work of the first author [13] yields the estimate

$$\begin{aligned} {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(\varphi , \varphi ) - {{\,\textrm{Rm}\,}}_g(\varphi , \varphi )&\ge 2 \lambda (-\chi ''(\lambda \rho )) S([\varphi ] d\rho , [\varphi ] d\rho ) \\&\quad + \varphi ^{ij} \varphi ^{kl} \chi '(\lambda \rho ) (-2 D_i D_k \rho \, S_{jl} - \frac{1}{2} \chi '(\lambda \rho ) |D\rho |_{{\hat{g}}_\lambda }^2 S_{ik} S_{jl}) \\&\quad - C \chi '(\lambda \rho ) |[\varphi ] d\rho | - C \lambda ^{-1} \end{aligned}$$

where $C>0$ is a positive constant independent of $\lambda$.

To pick up the sectional curvatures, we choose for $1 \le p,q \le n$ the two-form $\varphi _{pq}$ by specifying the components $(\varphi _{pq})^{ij} = \delta _p^i\delta _q^j - \delta _p^j\delta _q^i$. This implies the identities

$$\begin{aligned} {[}\varphi _{pq}]d\rho = \nabla _p\rho \ e_q - \nabla _q\rho \ e_p\quad \text {and}\quad 4{{\,\textrm{Rm}\,}}(e_p, e_q, e_p, e_q) = {{\,\textrm{Rm}\,}}(\varphi _{pq}, \varphi _{pq}) \end{aligned}$$

Recall that S is supported in U, so $D\rho$ and $D^2\rho$ are uniformly bounded with respect to the metric ${\hat{g}}_{\lambda }$. Thus, the above estimate implies after summation the inequality

$$\begin{aligned} \begin{aligned}&\sum _{p=1}^m\sum _{q=p+1}^n( {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(e_p, e_q, e_p, e_q) - {{\,\textrm{Rm}\,}}(e_p, e_q, e_p, e_q))\\&\quad \ge \frac{1}{2} \lambda (-\chi ''(\lambda \rho ))\ \sum _{p=1}^m\sum _{q=p+1}^n S\Big (\nabla _p\rho \ e_q - \nabla _q\rho \ e_p,\ \nabla _p\rho \ e_q - \nabla _q\rho \ e_p\Big ) \\&\qquad - C\chi '(\lambda \rho ) - C\lambda ^{-1} \end{aligned}\nonumber \\ \end{aligned}$$

(4)

in the region $\{\rho \ge e^{-\lambda ^2}\}$. Here the constant C is independent of $\lambda$, but it does depend on $S, g, \rho , N$.

By our assumption on the scalar-valued second fundamental forms and Proposition 3.1 we deduce

(with respect to the metric g) at each point on $\partial N$. This allows us to fix a small number $a>0$ such that

(with respect to the metric g) in a small neighborhood of $\partial N$ where $\rho \ge e^{-\lambda ^2}$. From the construction of the metric ${\hat{g}}_{\lambda }$ we deduce

(with respect to the metric ${\hat{g}}_{\lambda }$) in a small neighborhood of $\partial N$ where $\rho \ge e^{-\lambda ^2}$. Moreover, we observe that

For a positive constant we observe the estimate

(5)

in a small neighborhood of the boundary $\partial N$ where $\rho \ge e^{-\lambda ^2}$. Combining the estimate Eq. (4) and the estimate Eq. (5) we obtain the estimate

$$\begin{aligned} \begin{aligned}&\sum _{p=1}^m\sum _{q=p+1}^n( {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(e_p, e_q, e_p, e_q) - {{\,\textrm{Rm}\,}}(e_p, e_q, e_p, e_q)) \\&\quad \ge a C(n,m) \lambda (-\chi ''(\lambda \rho )) |\nabla \rho |^2 - C\chi '(\lambda \rho ) - C\lambda ^{-1} \end{aligned} \end{aligned}$$

(6)

in the region $\{\rho \ge e^{-\lambda ^2}\}$.

We split the region into two sub-regions as follows. Let us fix a real number $s_0\in [0,1)$ such that $C\chi '(s_0) < \frac{\epsilon }{2}$. By the construction of the cut-off function $\chi$, we have $\inf _{0\le s\le s_0} (-\chi ''(s)) > 0$. This implies in the region $\{e^{-\lambda ^2}\le \rho < s_0\lambda ^{-1}\}$ the estimate

$$\begin{aligned}&\inf _{e^{-\lambda ^2}\le \rho<s_0\lambda ^{-1}}\left( \sum _{p=1}^m\sum _{q=p+1}^n( {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(e_p, e_q, e_p, e_q) - {{\,\textrm{Rm}\,}}(e_p, e_q, e_p, e_q))\right) \\&\quad \ge a C(n,m) \lambda \, \inf _{e^{-\lambda ^2}\le \rho <s_0\lambda ^{-1}}\left( (-\chi ''(\lambda \rho )) |\nabla \rho |^2\right) - C - C\lambda ^{-1}\\&\quad \ge a C(n,m) \lambda \, \inf _{0\le s\le s_0} (-\chi ''(s)) - C - C\lambda ^{-1} \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \inf _{e^{-\lambda ^2}\le \rho <s_0\lambda ^{-1}}\left( \sum _{p=1}^m\sum _{q=p+1}^n( {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(e_p, e_q, e_p, e_q) - {{\,\textrm{Rm}\,}}(e_p, e_q, e_p, e_q))\right) \rightarrow \infty \end{aligned}$$

as $\lambda \rightarrow \infty$. Moreover, in the region $\{ \rho \ge s_0\lambda ^{-1}\}$ we have

$$\begin{aligned}&\inf _{\rho \ge s_0\lambda ^{-1}}\left( \sum _{p=1}^m\sum _{q=p+1}^n( {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(e_p, e_q, e_p, e_q) - {{\,\textrm{Rm}\,}}(e_p, e_q, e_p, e_q))\right) \\&\ge -C\chi '(s_0) - C\lambda ^{-1} \end{aligned}$$

Since $C\chi '(s_0) < \frac{\epsilon }{2}$, it follows that

$$\begin{aligned} \inf _{\rho \ge s_0\lambda ^{-1}}\left( \sum _{p=1}^m\sum _{q=p+1}^n( {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(e_p, e_q, e_p, e_q) - {{\,\textrm{Rm}\,}}(e_p, e_q, e_p, e_q))\right) \ge -\epsilon \end{aligned}$$

if $\lambda > 0$ is sufficiently large. Putting the above together, we conclude that

$$\begin{aligned} \inf _{\rho \ge e^{-\lambda ^2}}\left( \sum _{p=1}^m\sum _{q=p+1}^n( {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(e_p, e_q, e_p, e_q) - {{\,\textrm{Rm}\,}}(e_p, e_q, e_p, e_q))\right) \ge -\epsilon \end{aligned}$$

if $\lambda > 0$ is sufficiently large. This completes the proof of Proposition 4.1. $\square$

We next consider the region $\{\rho <e^{-\lambda ^2}\}$:

Proposition 4.2

(Curvature estimates in outer gluing region) Suppose that $h_g - h_{{\tilde{g}}}$ is m-positive on the boundary $\partial N$. Let $\epsilon > 0$ be an arbitrary positive real number. If $\lambda = \lambda (\epsilon , \beta ) >0$ is sufficiently large, then

$$\begin{aligned} \sum _{p=1}^m\sum _{q=p+1}^n ({{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(x)(e_p, e_q, e_p, e_q) -{{\,\textrm{Rm}\,}}_{{\tilde{g}}}(x)(e_p, e_q, e_p, e_q)) \ge -\epsilon \end{aligned}$$

for any ${\hat{g}}_{\lambda }$-orthonormal frame $\{e_1, \dots , e_n\}$ and any $x\in N$ in the region $\{\rho (x)< e^{-\lambda ^2}\}$.

Proof

In the region $\{\rho <e^{-\lambda ^2}\}$, we have ${\hat{g}}_{\lambda }={\tilde{g}}+{\tilde{h}}_{\lambda }$, where ${\tilde{h}}_{\lambda }$ is defined by

$$\begin{aligned} {\tilde{h}}_{\lambda } = -\lambda \rho ^2\beta (\lambda ^{-2}\log \rho )S \end{aligned}$$

Let $\{e_1, \dots , e_n\}$ be a geodesic normal frame around x with respect to the metric ${\hat{g}}_{\lambda }$. Equation (12) in work of the first author [13] implies

$$\begin{aligned} {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(\varphi , \varphi )-{{\,\textrm{Rm}\,}}_{{\tilde{g}}}(\varphi , \varphi ) \ge \ {}&2\lambda \beta (\lambda ^{-2}\log \rho )S([\varphi ]d\rho , [\varphi ]d\rho ) - L\lambda ^{-1} \end{aligned}$$

(7)

for any two-form $\varphi = \varphi ^{ij}e_i\wedge e_j$ and a positive constant $L > 0$ independent of $\lambda$. Choosing for $1 \le p,q \le n$ the two-form $\varphi _{pq}$ by $(\varphi _{pq})^{ij} = \delta _p^i\delta _q^j - \delta _p^j\delta _q^i$ in Eq. (7) and summing over p, q, we obtain

$$\begin{aligned}&\sum _{p=1}^m\sum _{q=p+1}^n( {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(e_p, e_q, e_p, e_q) - {{\,\textrm{Rm}\,}}_{{\tilde{g}}}(e_p, e_q, e_p, e_q)) \\&\quad \ge \frac{1}{2}\lambda \beta (\lambda ^{-2}\log \rho ) \sum _{p=1}^m\sum _{q=p+1}^n S\Big (\nabla _p\rho \ e_q - \nabla _q\rho \ e_p,\ \nabla _p\rho \ e_q - \nabla _q\rho \ e_p\Big ) - L\lambda ^{-1} \end{aligned}$$

Proceeding similarly as in the proof of Proposition 4.1, we have

$$\begin{aligned} \sum _{p=1}^m\sum _{q=p+1}^n S\Big (\nabla _p\rho \ e_q - \nabla _q\rho \ e_p,\ \nabla _p\rho \ e_q - \nabla _q\rho \ e_p\Big ) \ge 2a|\nabla \rho |_{{\tilde{g}}}^2 \end{aligned}$$

(with respect to the metric ${\hat{g}}_{\lambda }$) in a neighborhood of $\partial N$ where $\rho < e^{-\lambda ^2}$. This implies

$$\begin{aligned}&\sum _{p=1}^m\sum _{q=p+1}^n( {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(e_p, e_q, e_p, e_q) - {{\,\textrm{Rm}\,}}_{{\tilde{g}}}(e_p, e_q, e_p, e_q)) \ge a\lambda \beta (\lambda ^{-2}\log \rho ) |\nabla \rho |^2 - L\lambda ^{-1} \end{aligned}$$

(8)

in the region $\{\rho <e^{-\lambda ^2}\}$. Hence, if $\lambda >0$ is sufficiently large, then we have

$$\begin{aligned} \inf _{\rho <e^{-\lambda ^2}}\left( \sum _{p=1}^m\sum _{q=p+1}^n( {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(e_p, e_q, e_p, e_q) - {{\,\textrm{Rm}\,}}_{{\tilde{g}}}(e_p, e_q, e_p, e_q))\right) \ge -\epsilon \end{aligned}$$

From this, the assertion follows. $\square$

Combining Propositions 4.1 and 4.2, we can summarize the results in this section:

Corollary 4.3

Suppose that $h_g - h_{{\tilde{g}}}$ is m-positive on the boundary $\partial N$. Let $\epsilon >0$ be an arbitrary positive real number. If $\lambda = \lambda (\epsilon , \chi , \beta ) >0$ is sufficiently large, then we have the pointwise inequality

$$\begin{aligned}&\sum _{p=1}^m\sum _{q=p+1}^n {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(x)(e_p, e_q, e_p, e_q) \\&\quad \ge \min \left\{ \sum _{p=1}^m\sum _{q=p+1}^n {{\,\textrm{Rm}\,}}_g(x)(e_p, e_q, e_p, e_q), \sum _{p=1}^m\sum _{q=p+1}^n {{\,\textrm{Rm}\,}}_{{\tilde{g}}}(x)(e_p, e_q, e_p, e_q)\right\} - \epsilon \end{aligned}$$

for any ${\hat{g}}_{\lambda }$-orthonormal frame $\{e_1, \dots , e_n\}$ and any $x\in N$.

5 Proof of Theorem 1.2 on Preserving Positive Intermediate Curvature

Suppose that $h_g - h_{{\tilde{g}}}$ is m-positive on the boundary $\partial N$. Suppose also that (N, g) and $(N, {\tilde{g}})$ have positive m-intermediate curvature.

Fix a point $x \in N$ and let $\{E_1, \dots , E_N\}$ be an orthonormal basis of the tangent space $T_x N$ with respect to the Riemannian metric ${\hat{g}}_{\lambda }$. We want to show that

$$\begin{aligned} \sum _{p=1}^m\sum _{q=p+1}^n {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(E_p, E_q, E_p, E_q) > 0 \end{aligned}$$

We divide the proof into two cases: In the first case $x \in N$ is in the inner gluing region $\{\rho \ge e^{-\lambda ^2}\}$; and in the second case $x \in N$ is in the outer gluing region $\{\rho < e^{-\lambda ^2}\}$.

For the first case, we have $g = {\hat{g}}_{\lambda } - h_{\lambda }$ where $h_{\lambda }(x)=\lambda ^{-1}\chi (\lambda \rho )S(x)$. Fix a point $x\in N$. We evolve the orthonormal basis $\{E_1, \dots , E_n\}$ in the tangent space $T_x N$ by the linear ordinary differential equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d}{ds}E_i(s) &{}=\ \frac{1}{2}h_{\lambda } \circ E_i(s)\\ E_i(0)&{}=\ E_i \end{array}\right. } \end{aligned}$$

(9)

Then we see that the basis $\{E_i(s)\}$ remains orthonormal with respect to the metric $g_s= {\hat{g}}_{\lambda }-sh_{\lambda }$. We define $e_i:= E_i(1)$. Then the basis $\{e_1, \dots , e_n\}$ is orthonormal with respect to the metric $g = {\hat{g}}_{\lambda } - h_{\lambda }$. It follows that

$$\begin{aligned} \sum _{p=1}^m\sum _{q=p+1}^n {{\,\textrm{Rm}\,}}_{g}(e_p, e_q, e_p, e_q) > 0 \end{aligned}$$

In view of Corollary 4.3, it suffices to show that

$$\begin{aligned} \sum _{p=1}^m\sum _{q=p+1}^n {{\,\textrm{Rm}\,}}_{g}(E_p, E_q, E_p, E_q) > 0 \end{aligned}$$

By writing $E_i(s)=A_i^k(s)E_k$, we observe that

$$\begin{aligned} \frac{d}{ds}E_i(s)&= \frac{d}{ds}A_i^k(s)E_k = \frac{1}{2}h_l^kA_i^l(s)E_k = \frac{1}{2}\lambda ^{-1}\chi (\lambda \rho )S_l^kA_i^l(s)E_k \end{aligned}$$

for any $s\in [0,1]$. This implies the estimate

$$\begin{aligned}&\Big |\frac{d}{ds}{{\,\textrm{Rm}\,}}_g(E_p(s), E_q(s), E_p(s), E_q(s))\Big | \le C\lambda ^{-1} \end{aligned}$$

where C is a positive constant depending only on $N,g,{\tilde{g}}$ and $\chi$. This implies that

$$\begin{aligned} \Big |{{\,\textrm{Rm}\,}}_g(E_p, E_q, E_p, E_q) - {{\,\textrm{Rm}\,}}_g(e_p, e_q, e_p, e_q)\Big |&\le \int _0^1\frac{C}{\lambda }d\tau \le \frac{C}{\lambda } \end{aligned}$$

Hence we have

$$\begin{aligned} \sum _{p=1}^m\sum _{q=p+1}^n {{\,\textrm{Rm}\,}}_{g}(E_p, E_q, E_p, E_q) \ge \frac{1}{2} \sum _{p=1}^m\sum _{q=p+1}^n {{\,\textrm{Rm}\,}}_{g}(e_p, e_q, e_p, e_q) > 0 \end{aligned}$$

if $\lambda >0$ is sufficiently large.

For the second case, we have ${\tilde{g}} = {\hat{g}}_{\lambda } + \lambda \rho ^2\beta (\lambda ^{-2}\log \rho )S$. Thus $h_{\lambda } = \lambda \rho ^2\beta (\lambda ^{-2}\log \rho )S$. Following the same argument as in the first case and using the fact that

$$\begin{aligned} |h_{\lambda }|\le C(g,{\tilde{g}},\beta )e^{-\lambda ^2}\le \frac{C(g,{\tilde{g}},\beta )}{\lambda } \end{aligned}$$

in the region $\{\rho <e^{-\lambda ^2}\}$ for sufficiently large $\lambda >0$, we also deduce

$$\begin{aligned} \sum _{p=1}^m\sum _{q=p+1}^n {{\,\textrm{Rm}\,}}_{{\tilde{g}}}(E_p, E_q, E_p, E_q) \ge \frac{1}{2} \sum _{p=1}^m\sum _{q=p+1}^n {{\,\textrm{Rm}\,}}_{{\tilde{g}}}({\tilde{e}}_p, {\tilde{e}}_q, {\tilde{e}}_p, {\tilde{e}}_q) > 0 \end{aligned}$$

for sufficiently large $\lambda > 0$.

Combining the two cases together, Corollary 4.3 implies that

$$\begin{aligned} \sum _{p=1}^m\sum _{q=p+1}^n {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}(E_p, E_q, E_p, E_q) >0 \end{aligned}$$

for sufficiently large $\lambda >0$. Since the point $x\in N$ and the orthonormal basis $\{E_1,...,E_n\}\subset T_x N$ are arbitrary, we conclude that

$$\begin{aligned} {{\,\textrm{Rm}\,}}_{{\hat{g}}_{\lambda }}\in {{\,\textrm{Int}\,}}({\mathcal {C}}_m) \end{aligned}$$

on N for sufficiently large $\lambda > 0$. This finishes the proof of the theorem.

6 Proof of Theorem 1.5 on Boundaries of Partially Torical Bands

In this section, we prove Theorem 1.5 by using the strategy outlined by Gromov and Lawson. We prove a generalization of their doubling lemma (as stated in Lemma 1.3).

Once we have established the doubling lemma Theorem 1.5 follows directly by the generalization of the Geroch conjecture, Theorem 1.1.

Lemma 6.1

(Doubling for positive $m-$intermediate curvature) Suppose $(N^n,g)$ is an orientable compact smooth Riemannian manifold with smooth boundary $\partial N$, such that the metric g has positive m-intermediate curvature, and the boundary $\partial N$ is strictly m-convex (possibly finitely many connected components). Then, the double of N carries a metric of positive m-intermediate curvature.

We mimic the doubling process by Gromov and Lawson, see Theorem 5.7 in [8]. We will closely follow their notations and constructions. For completeness we describe their construction in detail.

We first shrink the manifold N a little bit while preserving its boundary condition as follow: let $N_1=N\backslash C$ where C is a thin collar of the boundary $\partial N$ and $N_1$ is chosen so that $\partial N_1$ is still strictly m-mean convex. We then consider the Riemannian product $N\times I$ with Riemannian metric $g_N+dt\otimes dt$ and define $D(N) = \{p \in N\times I \mid \text {dist}(p, N_1) = \epsilon \}$, where $0 <\epsilon \ll 1$. Note that D(N) is homeomorphic to the double of N.

Now, we fix a point $x\in \partial N_1$, and let $\sigma$ be the geodesic segment in $N_1$ emanating orthogonally from $\partial N_1$ at x. Then the product $\sigma \times I$ will be totally geodesic in the product $N\times I$.

Let $\mu _1,...\mu _{n-1}$ be the principal curvatures of $\partial N_1$ at x. By the construction of M. Gromov and H.-B. Lawson the principle curvatures of D(N) will be of the following form for a point $x_{\theta }$ corresponding to an angle $\theta$ (see Fig. 8 in [8]):

$$\begin{aligned} \lambda _k=(\mu _k+O(\epsilon ))\cos \theta +O(\epsilon ^2 ) \; \text {for} \; k = 1, \dots , n-1, \; \text {and} \;\lambda _n=\frac{1}{\epsilon }\cos \theta +O(\epsilon ) \end{aligned}$$

(10)

As in the Fig. 8 in [8], we have a natural polar coordinates describing these $x_\theta$, let us denote $pr_N(x_\theta )$ the projection of $x_\theta$ to the corresponding point on $\partial N_1$. Since the bilinear forms $h_{\partial N_1}$ at $pr_N(x_\theta )$ and $h_{D(N)}$ at $x_{\theta }$ are diagonalized simultaneously in this construction, we have the following relation:

$$\begin{aligned} h_{D(N)}\Big |_{x_\theta }=\left( (\cos \theta ) h_{\partial N_1}\Big |_{pr_N(x_\theta )} +O(\epsilon )\right) \oplus \left( \frac{1}{\epsilon }\cos \theta +O(\epsilon )\right) (\nu ^\flat \otimes \nu ^\flat )\Big |_{pr_N(x_\theta )} \end{aligned}$$

We apply the Gauss equation to D(N) as a submanifold of $N\times I$ at the point $x_\theta$: We have for any orthonormal basis $\{e_i\}_{i=1}^n$ at the point $x_\theta$ the relation

$$\begin{aligned} \text {Rm}^{D(N)}(e_i,e_j,e_i,e_j)&= \text {Rm}^{N\times I}(e_i,e_j,e_i,e_j) \nonumber \\&\quad +h_{D(N)}(e_i,e_i)h_{D(N)}(e_j,e_j)-h_{D(N)}(e_i,e_j)^2 \end{aligned}$$

(11)

We note that the second fundamental form terms of D(N) are related to the second fundamental form terms of the boundary $\partial N_1$. This implies

where the terms of order $O(\frac{1}{\epsilon ^2})$ cancelled.

Summation of Eq. (11) yields the following identity for the m-intermediate curvature:

The result then follows from Proposition 3.1 by choosing $\epsilon >0$ sufficient small.