Spectral splitting theorem and ends of minimal hypersurfaces

Abstract: In this paper, we give a new proof of the splitting theorem on manifolds with nonnegative spectral Ricci curvature proved in [APX24, CMMR24, HW26]. Furthermore, by constructing weighted minimizing geodesics at infinity, we show that minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature must have finite ends, generalizing the result of Li-Wang [LW04] on manifolds with nonnegative sectional curvature.

• The paper establishes spectral splitting from a weighted minimizing line when α < 4/(n−1), using weighted Laplacian comparison, Busemann functions, and Bochner rigidity to prove an isometric ℝ-factor.
• It proves that finite-index minimal hypersurfaces in manifolds with nonnegative biRic curvature have finitely many ends for ambient dimension n ≤ 5, without assuming the immersion is proper.
• It develops a geodesic-lines-at-infinity theorem showing that infinitely many ends under nonnegative spectral Ricci curvature outside a compact set produce a weighted minimizing line, enabling the minimal-hypersurface application.

Overview

This paper, by Han Hong and Gaoming Wang, makes two contributions to the geometry of noncompact Riemannian manifolds under integral (spectral) curvature conditions. First, it gives a new proof of the splitting theorem for manifolds with nonnegative spectral Ricci curvature, adapting the classical Cheeger–Gromoll Busemann function argument (2605.14931). Second, and more substantially, it proves that finite-index minimal hypersurfaces in manifolds with nonnegative biRic curvature have finitely many ends, extending a theorem of Li–Wang from nonnegative sectional curvature to the biRic setting and removing the properness assumption on the immersion.

Background and motivation

For a noncompact $n$-manifold $(M,g)$, the paper defines $\Ric(x) = \inf\{\Ric(v,v) : g(v,v)=1\}$ and says $M$ has nonnegative spectral $\alpha$-Ricci curvature, $\alpha \geq 0$, if $\lambda_1(-\alpha\Delta + \Ric) \geq 0$, equivalently $\int_M \alpha|\nabla\varphi|^2 + \Ric\,\varphi^2 \geq 0$ for all compactly supported $\varphi$. By elliptic theory this is equivalent to the existence of a positive $u \in C^{2,\beta}(M)$ solving $(M,g)$0, with Neumann condition on $(M,g)$1 if present.

The spectral splitting theorem of Antonelli–Pozzetta–Xu and, independently, Catino–Mari–Mastrolia–Roncoroni states that if $(M,g)$2, $(M,g)$3, and $(M,g)$4 has at least two ends, then $(M,g)$5 pointwise and $(M,g)$6 splits isometrically as $(M,g)$7 with $(M,g)$8 closed of nonnegative Ricci curvature. Antonelli–Pozzetta–Xu used $(M,g)$9-bubbles, Catino–Mari–Mastrolia–Roncoroni used criticality theory of Schrödinger operators, and Hong–Wang previously treated the boundary case via the second variation of weighted length and Liu's deformation argument. A notable feature emphasized in the paper is that the two-ends hypothesis cannot be weakened to the existence of a minimizing line: Antonelli–Pozzetta–Xu constructed a counterexample.

The topological motivation comes from stable minimal hypersurface theory. If the ambient manifold $\Ric(x) = \inf\{\Ric(v,v) : g(v,v)=1\}$0 has biRic $\Ric(x) = \inf\{\Ric(v,v) : g(v,v)=1\}$1, then combining the Gauss equation with the stability inequality yields

$\Ric(x) = \inf\{\Ric(v,v) : g(v,v)=1\}$2

so every stable minimal hypersurface in $\Ric(x) = \inf\{\Ric(v,v) : g(v,v)=1\}$3 carries nonnegative spectral Ricci curvature — an observation due to Shen–Ye generalizing the Fischer-Colbrie–Schoen theorem. Prior results on ends: complete stable minimal hypersurfaces in $\Ric(x) = \inf\{\Ric(v,v) : g(v,v)=1\}$4 have one end (Cao–Shen–Zhu); finite-index minimal hypersurfaces in $\Ric(x) = \inf\{\Ric(v,v) : g(v,v)=1\}$5 have finitely many ends (Li–Wang); and properly immersed finite-index minimal hypersurfaces in nonnegatively curved manifolds have finitely many ends (Li–Wang). The Li–Wang argument relies on harmonicity of Busemann functions under nonnegative sectional curvature and only detects ends extending to infinity, which forces the properness assumption. The paper removes both restrictions.

Main results

The paper proves three statements. Theorem 1 (a strengthening of prior splitting theorems): if $\Ric(x) = \inf\{\Ric(v,v) : g(v,v)=1\}$6 and $\Ric(x) = \inf\{\Ric(v,v) : g(v,v)=1\}$7 admits a weighted minimizing line with respect to $\Ric(x) = \inf\{\Ric(v,v) : g(v,v)=1\}$8, then $\Ric(x) = \inf\{\Ric(v,v) : g(v,v)=1\}$9 and $M$0 splits off an $M$1 factor. Since a weighted minimizing line always exists when $M$2 has at least two ends, this recovers the two-ends theorem while weakening its hypothesis. Note the paper remarks that the weighted length of the line may be finite.

Theorem 2 is the principal new result: if $M$3 has nonnegative biRic curvature and $M$4, then every finite-index minimal hypersurface in $M$5 has finitely many ends. No properness of the immersion is required, and the sectional curvature hypothesis of Li–Wang is replaced by the strictly weaker biRic condition. The dimensional restriction $M$6 enters through the ambient curvature assumption in the Bernstein-type framework (consistent with the recent stable Bernstein theorems of Chodosh–Li–Stryker and Mazet).

Theorem 3, the analytic engine behind Theorem 2, is of independent interest: if the spectral $M$7-Ricci curvature of $M$8 is nonnegative on $M$9 for some compact $\alpha$0, with $\alpha$1, and $\alpha$2 has infinitely many ends, then $\alpha$3 contains a weighted minimizing geodesic line outside a larger compact set. This is a spectral analogue of a result of Bi–Zhu on manifolds with nonnegative Ricci curvature outside a compact set, proved concurrently using the same geodesic-lines-at-infinity technique.

The spectral splitting argument

The new proof of the splitting theorem follows the Cheeger–Gromoll blueprint. The key technical input is a spectral Laplacian comparison: for $\alpha$4 with $\alpha$5 and $\alpha$6, the weighted distance satisfies, in the viscosity sense,

$\alpha$7

The proof applies the second variation formula for weighted length with test vector fields $\alpha$8 along a minimizing geodesic, substitutes $\alpha$9, and uses the equation for $\alpha \geq 0$0 together with the constraint $\alpha \geq 0$1 to absorb the gradient terms involving $\alpha \geq 0$2 and $\alpha \geq 0$3 into a constant multiple of $\alpha \geq 0$4. The case where $\alpha \geq 0$5 and $\alpha \geq 0$6 are joined by a chain of minimizing rays and lines is handled by a limiting argument, sending an intermediate point on an intermediate line to infinity.

With this comparison in hand, the weighted Busemann functions $\alpha \geq 0$7 are constructed as monotone, uniformly bounded, Lipschitz limits of $\alpha \geq 0$8. Passing to the limit in the weak formulation using the Laplacian comparison shows $\alpha \geq 0$9; since the triangle inequality gives $\lambda_1(-\alpha\Delta + \Ric) \geq 0$0 with equality along the line, the maximum principle yields $\lambda_1(-\alpha\Delta + \Ric) \geq 0$1 and $\lambda_1(-\alpha\Delta + \Ric) \geq 0$2, hence smoothness by elliptic regularity. The rigidity step combines the Bochner formula with the identity $\lambda_1(-\alpha\Delta + \Ric) \geq 0$3 to obtain

$\lambda_1(-\alpha\Delta + \Ric) \geq 0$4

and Kato's inequality $\lambda_1(-\alpha\Delta + \Ric) \geq 0$5. Since $\lambda_1(-\alpha\Delta + \Ric) \geq 0$6 precisely when $\lambda_1(-\alpha\Delta + \Ric) \geq 0$7, one concludes $\lambda_1(-\alpha\Delta + \Ric) \geq 0$8, $\lambda_1(-\alpha\Delta + \Ric) \geq 0$9 is constant, and the splitting follows. It is worth noting that the rigidity step requires the sharper threshold $\int_M \alpha|\nabla\varphi|^2 + \Ric\,\varphi^2 \geq 0$0, while the Laplacian comparison only needs $\int_M \alpha|\nabla\varphi|^2 + \Ric\,\varphi^2 \geq 0$1; the paper states the theorem under the latter hypothesis, and the proof implicitly uses the former in the Bochner step. Antonelli–Li–Sweeney have since improved the threshold to $\int_M \alpha|\nabla\varphi|^2 + \Ric\,\varphi^2 \geq 0$2 under an additional compact boundary component.

Constructing weighted minimizing lines at infinity

The existence of a weighted minimizing line when $\int_M \alpha|\nabla\varphi|^2 + \Ric\,\varphi^2 \geq 0$3 has at least two ends is proved by an exhaustion argument: one minimizes $\int_M \alpha|\nabla\varphi|^2 + \Ric\,\varphi^2 \geq 0$4 for a perturbed weight $\int_M \alpha|\nabla\varphi|^2 + \Ric\,\varphi^2 \geq 0$5 (equal to $\int_M \alpha|\nabla\varphi|^2 + \Ric\,\varphi^2 \geq 0$6 on $\int_M \alpha|\nabla\varphi|^2 + \Ric\,\varphi^2 \geq 0$7, growing outside $\int_M \alpha|\nabla\varphi|^2 + \Ric\,\varphi^2 \geq 0$8) among curves joining two unbounded end components, and takes $\int_M \alpha|\nabla\varphi|^2 + \Ric\,\varphi^2 \geq 0$9; a smooth locally minimizing limit component of infinite $\varphi$0-length is the desired line.

For the ends theorem, the argument is more delicate because $\varphi$1 need not be mean convex, so free-boundary rays (the tool of the earlier Hong–Wang theorem) are unavailable and lines are required instead. The proof of Theorem 3 proceeds as follows. If $\varphi$2 has infinitely many ends, some unbounded component $\varphi$3 of $\varphi$4 itself has infinitely many ends, and by a proposition of Hong–Wang each interior end carries a free-boundary $\varphi$5-minimizing geodesic ray perpendicular to $\varphi$6. Since the footpoints $\varphi$7 on the compact $\varphi$8 accumulate, a subsequence of rays converges smoothly to a ray $\varphi$9. Given any compact $u \in C^{2,\beta}(M)$0 and $u \in C^{2,\beta}(M)$1, one picks $u \in C^{2,\beta}(M)$2 large enough that $u \in C^{2,\beta}(M)$3 is $u \in C^{2,\beta}(M)$4-close to $u \in C^{2,\beta}(M)$5 on $u \in C^{2,\beta}(M)$6 in weighted length. Connecting far-out points of $u \in C^{2,\beta}(M)$7 and $u \in C^{2,\beta}(M)$8 by a weighted minimizer $u \in C^{2,\beta}(M)$9, a triangle-inequality estimate shows $(M,g)$00 cannot touch $(M,g)$01 provided $(M,g)$02; letting $(M,g)$03 yields a smooth $(M,g)$04-minimizing line outside $(M,g)$05 (possibly with several components, each a line).

The contradiction is then obtained by perturbing $(M,g)$06 (via the local argument of Antonelli–Pozzetta–Xu) to achieve the strict inequality $(M,g)$07 on $(M,g)$08. The second variation formula for $(M,g)$09 along the line, with $(M,g)$10, gives an inequality whose standard test function forces $(M,g)$11 along $(M,g)$12 — contradicting strict positivity. Applying this to $(M,g)$13, where stability outside $(M,g)$14 and the Shen–Ye inequality give nonnegative spectral $(M,g)$15-Ricci curvature, proves Theorem 2.

Limitations and open questions

Several restrictions are intrinsic to the method. The threshold $(M,g)$16 (and the sharper $(M,g)$17 in the Bochner rigidity step) is not addressed here, and the counterexample of Antonelli–Pozzetta–Xu shows that some hypothesis beyond a minimizing line is needed in general, so the exact optimal condition on $(M,g)$18 and on the line hypothesis remains open. The dimensional bound $(M,g)$19 in Theorem 2 is inherited from the current state of stable Bernstein theory; whether the ends-finiteness holds in all dimensions, or whether biRic can be weakened further (for instance to scalar curvature in low dimensions), is not settled. The paper also leaves open whether the strict perturbation of $(M,g)$20 can be arranged with control that would yield quantitative rigidity rather than a pure contradiction. Finally, the concurrent and independent Bi–Zhu result suggests that the geodesic-lines-at-infinity technique extends to other curvature-at-infinity hypotheses; a unified treatment is not attempted here.

Conclusion

The paper contributes a fourth proof of the spectral splitting theorem, via weighted Busemann functions and a spectral Laplacian comparison, replacing the two-ends hypothesis by the existence of a weighted minimizing line. Its main application shows that finite-index minimal hypersurfaces in manifolds of nonnegative biRic curvature, $(M,g)$21, have finitely many ends — a genuine generalization of Li–Wang that drops properness and weakens nonnegative sectional curvature to biRic, with the spectral condition at infinity supplying the needed weighted minimizing line. The result ties the topology of minimal hypersurface ends to the spectral geometry of Schrödinger operators, in line with the recent program of stable Bernstein theorems.