Connected sum of manifolds with spectral Ricci lower bounds

Technical Summary: Connected Sum of Manifolds with Spectral Ricci Lower Bounds

Problem Statement
The paper addresses the stability of spectral Ricci lower bounds under the topological operation of connected sums. Specifically, it investigates whether two smooth $n$-manifolds ($n \geq 3$) admitting a complete Riemannian metric satisfying a spectral condition of the form $\lambda_1(-\gamma\Delta + \text{Ric}) \geq \lambda$ can be connected to form a new manifold $M \# N$ that also admits a metric satisfying the same condition.

The condition $\lambda_1(-\gamma\Delta + \text{Ric}) \geq \lambda$ is defined as the existence of a positive function $u$ such that $-\gamma\Delta u + \text{Ric} \cdot u \geq \lambda u$, or equivalently, the validity of the inequality $\int_M (\gamma|\nabla \phi|^2 + \text{Ric} \cdot \phi^2) \geq \lambda \int_M \phi^2$ for all compactly supported $C^1$ functions. This is a global, spectral condition on the Ricci curvature, which is generally weaker than pointwise lower bounds.

A critical threshold for the parameter $\gamma$ is identified as $\frac{n-1}{n-2}$. Previous work (notably Bour–Carron [4]) established volume and Betti number bounds for manifolds where $\gamma \leq \frac{n-1}{n-2}$. However, in the "supercritical" range $\gamma > \frac{n-1}{n-2}$, counterexamples to these bounds exist (e.g., via warped products on $S^{n-1} \times S^1$). The authors aim to demonstrate that the geometric and topological behavior of manifolds in this supercritical range differs significantly, specifically by proving that the spectral condition is preserved under connected sums when $\gamma > \frac{n-1}{n-2}$.

Methodology
The proof employs a geometric construction analogous to the Gromov–Lawson and Schoen–Yau techniques used for scalar curvature, but adapted for the spectral Ricci condition. The core strategy involves:

1. Local Model Construction: The authors utilize a "toy model" inspired by Bour–Carron [4], involving a warped product metric $dr^2 + f(r)^2 g_{S^{n-1}}$ and a specific function $u = f(r)^{2-n}$. They observe that for an arbitrary convex warping function $f$, this pair satisfies a specific differential identity. By choosing $f(r) \approx |r|$ for large $|r|$, they construct a local metric on $\mathbb{R}^n \# \mathbb{R}^n$ that satisfies the spectral condition with $\lambda=0$.

2. Globalization via Green's Functions: To extend this local model to a general manifold $M$, the authors remove two small balls $B(x_1, \epsilon)$ and $B(x_2, \epsilon)$ and glue in a "tunnel" $S^{n-1} \times [0, 1]$.

   • Instead of merely merging metrics, they must merge the associated eigenfunctions.
   • The authors observe that near the singularity, the model function behaves like $r^{2-n}$, which corresponds to the Green's function of the Laplacian.
   • Consequently, they solve the equation $-\gamma\Delta u + \text{Ric} \cdot u = (\lambda - \epsilon)u + (\delta_{x_1} + \delta_{x_2})$ on the original manifold $M$ to obtain a Green's function $u$ with poles at $x_1$ and $x_2$.
   • The new eigenfunction $\hat{u}$ on the connected sum is constructed by smoothly interpolating between the solution $u$ on the original manifold and the model solution $f^{2-n}$ within the tunnel.

3. Analytic Estimates: The proof relies on precise asymptotic estimates of the metric and the Green's function near the poles (Lemma 1, Lemma 7, Lemma 8). The authors define a cutoff function to blend the original metric $g$ with the warped product metric in the tunnel region. They then compute the Ricci curvature and the Laplacian of the new eigenfunction in two regions:

   • Region I (Near the original manifold): Where the metric is close to the original.
   • Region II (The center of the tunnel): Where the metric is the warped product.
     The crucial step involves showing that the term $\gamma(n-2) - (n-1)$, which arises from the interaction of the Laplacian and Ricci terms, is positive. This positivity is guaranteed only if $\gamma > \frac{n-1}{n-2}$.

Key Contributions and Results
The primary result is Theorem 1, which states:
Let $n \geq 3$, $\gamma > \frac{n-1}{n-2}$, and $\lambda \in \mathbb{R}$. If $(M, g)$ is a complete manifold satisfying $\lambda_1(-\gamma\Delta + \text{Ric}) \geq \lambda$, then for any two points $x_1, x_2 \in M$, there exists a complete manifold $M'$ (topologically $M$ with a tunnel connecting $x_1$ and $x_2$) and a metric $g'$ such that:

1. $g'$ coincides with $g$ outside small neighborhoods of $x_1$ and $x_2$.
2. $M'$ satisfies $\lambda_1(-\gamma\Delta_{g'} + \text{Ric}_{g'}) \geq \lambda - \epsilon$ for any $\epsilon > 0$.

Corollaries:

• Corollary 1: The connected sum $M_1 \# M_2$ of two such manifolds admits a complete metric satisfying the same spectral lower bound.
• Corollary 2: The connected sum $M \# (S^{n-1} \times S^1)$ admits such a metric.
• Sharpness of the Threshold: The authors demonstrate that the condition $\gamma > \frac{n-1}{n-2}$ is sharp. In the range $\gamma \leq \frac{n-1}{n-2}$, the connected sum operation would violate known topological obstructions (such as the Betti number bound $b_1(M) \leq n$ from Bour–Carron [4]). Specifically, the construction implies that for $\gamma > \frac{n-1}{n-2}$, one can construct manifolds with arbitrarily large first Betti numbers (e.g., $\#_k(S^{n-1} \times S^1)$) satisfying $\lambda_1(-\gamma\Delta + \text{Ric}) > 0$, thereby showing the Betti number bound fails in this regime.

Significance
The paper provides further evidence that the spectral Ricci lower bound undergoes a qualitative transition at the critical value $\gamma = \frac{n-1}{n-2}$.

• Below the threshold ($\gamma \leq \frac{n-1}{n-2}$): The condition imposes strong topological constraints (volume bounds, Betti number bounds).
• Above the threshold ($\gamma > \frac{n-1}{n-2}$): The condition is flexible enough to allow for topological operations like connected sums, which typically increase topological complexity (e.g., increasing $b_1$).

The authors note that their construction is technically distinct from the classical Gromov–Lawson surgery for scalar curvature. In the scalar curvature case, the warping factor must satisfy a specific differential inequality. In contrast, for the spectral Ricci condition in the supercritical range, the warping factor $f$ can be any convex function, provided it satisfies certain boundary conditions. This suggests a greater degree of freedom in constructing metrics with spectral Ricci lower bounds compared to pointwise bounds or scalar curvature bounds.

The paper concludes by noting that the result covers the codimension $k=n$ case of surgery, and raises the question of whether similar stability holds for surgeries of codimension $k$ where $2 < k < n$, provided $\gamma > \frac{k-1}{k-2}$.