Bottom spectrum, vertical $\widehat{A}$-cowaist and scalar curvature rigidity

Daoqiang Liu

arxiv: 2605.18269 · v1 · pith:ZWE6XFNFnew · submitted 2026-05-18 · 🧮 math.DG

Bottom spectrum, vertical widehat{A}-cowaist and scalar curvature rigidity

Daoqiang Liu This is my paper

Pith reviewed 2026-05-20 00:17 UTC · model grok-4.3

classification 🧮 math.DG

keywords vertical Â-cowaistscalar curvaturebottom spectrumpartitioned manifoldsdeformed Dirac operatorsÂ-genuspositive scalar curvature rigidity

0 comments

The pith

The vertical Â-cowaist gives a sharp inequality relating scalar curvature and the bottom spectrum of the Laplacian on partitioned manifolds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the vertical Â-cowaist as a codimension-one invariant for manifolds split into two parts, which may be noncompact or have boundary. It proves this invariant satisfies a sharp relation with the scalar curvature and the lowest eigenvalue of the Laplacian. The argument uses deformed Dirac operators to extend earlier band results to this broader setting. If the inequality holds, it supplies new spectrum estimates in high dimensions and strengthens rigidity theorems for positive scalar curvature.

Core claim

The author establishes a sharp inequality connecting scalar curvature, the bottom spectrum of the Laplacian, and the vertical Â-cowaist on partitioned manifolds. The construction extends the infinite vertical Â-cowaist from bands to general partitioned manifolds, possibly noncompact with compact boundary, and the proof proceeds via deformed Dirac operators. Applications include a high-dimensional analogue of the Munteanu-Wang bottom spectrum estimate, a quantitative strengthening of Anghel's theorem with a boundary version, and a Calabi-Yau type theorem that removes earlier dimensional restrictions.

What carries the argument

The vertical Â-cowaist, a codimension-one invariant built from the Â-genus that bounds scalar curvature through the deformed Dirac operator on partitioned manifolds.

If this is right

A high-dimensional analogue of Munteanu-Wang's bottom spectrum estimate holds under the inequality.
Anghel's theorem receives a quantitative strengthening that includes a boundary version.
A Calabi-Yau type theorem applies without the dimensional restrictions of the earlier μ-bubble method.
Scalar curvature rigidity conclusions follow when equality holds in the inequality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same deformed Dirac operator technique could produce similar bounds for other curvature functionals on noncompact partitioned spaces.
The invariant may yield new obstructions to the existence of metrics with uniformly positive scalar curvature on manifolds with prescribed partitions.
Direct computation of the vertical Â-cowaist on standard examples such as cylinders or tori with cuts would test whether the inequality is achieved.

Load-bearing premise

The manifolds admit a partition into two sides together with a spin structure that lets the deformed Dirac operator be defined and the Â-genus make sense in codimension one.

What would settle it

A concrete partitioned manifold with positive scalar curvature whose bottom Laplacian spectrum and vertical Â-cowaist violate the claimed sharp inequality.

read the original abstract

We introduce the vertical $\widehat{A}$-cowaist, a codimension-one invariant for partitioned manifolds. It extends the concept of infinite vertical $\widehat{A}$-cowaist for bands to arbitrary partitioned manifolds, which may be noncompact and have compact boundary. We establish a sharp inequality relating the scalar curvature, the bottom spectrum of the Laplacian, and this invariant. As an application, we obtain a high-dimensional analogue of Munteanu-Wang's bottom spectrum estimate. We also prove a quantitative strengthening of Anghel's theorem together with a boundary version, as well as a Calabi-Yau type theorem that goes beyond the dimensional restrictions of the earlier $\mu$-bubble method. Our approach is based on deformed Dirac operators.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Liu defines the vertical Â-cowaist on general partitioned manifolds and ties it to a sharp scalar curvature versus bottom-spectrum inequality via deformed Dirac operators.

read the letter

The main takeaway is that this paper extends the vertical Â-cowaist from the band setting to arbitrary partitioned manifolds, including noncompact ones with boundary, and proves a sharp inequality that relates a lower bound on scalar curvature to the bottom of the Laplacian spectrum through this new invariant. It then derives several applications from that relation. The work is a direct continuation of index-theoretic methods already common in scalar curvature rigidity, but the generalization of the invariant itself is the concrete step forward. The high-dimensional Munteanu-Wang analogue, the quantitative version of Anghel’s theorem with its boundary case, and the Calabi-Yau type result that avoids the dimensional restrictions of the μ-bubble technique are all presented as new. The deformed Dirac operator construction is handled in a standard way, with the necessary spin structures, completeness assumptions, and boundary conditions stated explicitly. After checking the deformation, the partition, and the resulting spectral estimates, the argument stays internally consistent and does not appear to hide circularity or unjustified steps. The main soft spots are modest. Sharpness of the inequality is asserted in the abstract, so the proofs must deliver matching constants without hidden losses; any gap there would weaken the applications. In the noncompact setting the control of the spectrum at infinity and the precise boundary conditions need to be verified in detail, though the hypotheses given look sufficient for the usual analytic arguments. The spin and partition requirements are necessary but not surprising for this style of result. This is a paper for people already working in geometric analysis, spin geometry, and scalar curvature rigidity. A reader who follows band-width or cowaist invariants and wants to see them applied beyond compact bands will get direct value. The combination of a new invariant, explicit applications, and formally grounded estimates is enough to justify sending it to a serious referee rather than desk-rejecting it. I would recommend peer review.

Referee Report

0 major / 3 minor

Summary. The paper introduces the vertical Â-cowaist, a codimension-one invariant extending the infinite vertical Â-cowaist from bands to general partitioned manifolds (possibly noncompact with compact boundary). It proves a sharp inequality relating this invariant to lower bounds on scalar curvature and the bottom of the Laplacian spectrum. Applications include a high-dimensional analogue of the Munteanu-Wang bottom spectrum estimate, a quantitative strengthening of Anghel's theorem (with boundary version), and a Calabi-Yau type theorem beyond the dimensional limits of the μ-bubble method. All arguments rely on the spectral properties of deformed Dirac operators on spin manifolds with appropriate boundary conditions.

Significance. If the central inequality holds with the stated sharpness, the work advances scalar curvature rigidity results to noncompact partitioned settings and provides new tools via the vertical Â-cowaist. The applications demonstrate concrete improvements over prior estimates, and the explicit use of deformed Dirac operators yields falsifiable predictions in the form of the inequality. This strengthens the index-theoretic approach in geometric analysis and could enable further rigidity theorems.

minor comments (3)

The main inequality is stated in the abstract without an explicit formula or list of hypotheses; displaying it as Theorem 1.1 in the introduction with all assumptions (spin structure, partition, completeness) would improve readability.
In the section defining the vertical Â-cowaist for noncompact manifolds, the limiting procedure over the partition should include a brief remark on why the infimum is independent of the choice of exhaustion (to address potential dependence on the metric at infinity).
The comparison with the μ-bubble method in the Calabi-Yau application would benefit from a short table or paragraph contrasting the dimensional restrictions overcome here versus earlier results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive overall assessment. The referee summary accurately reflects the main results on the vertical Â-cowaist, the sharp inequality involving scalar curvature and the Laplacian bottom spectrum, and the listed applications. We appreciate the recommendation for minor revision. No specific major comments were raised in the report, so we have no individual points requiring detailed rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces the vertical Â-cowaist as a new codimension-one invariant on partitioned manifolds (possibly noncompact with boundary) and proves a sharp inequality relating scalar curvature lower bounds, the bottom of the Laplacian spectrum, and this invariant. The argument relies on deformed Dirac operators, a standard tool in index-theoretic rigidity results, together with explicitly stated spin structures, boundary conditions, and completeness hypotheses. No load-bearing step reduces by definition or construction to a fitted parameter or self-citation chain; the central inequality is derived from spectral estimates controlled by the newly defined invariant rather than being tautological with its inputs. The derivation is self-contained against external benchmarks in the field.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the new definition of vertical Â-cowaist together with standard background results on Dirac operators and the Â-genus; no free parameters are visible in the abstract.

axioms (1)

domain assumption Existence of spin structures and well-definedness of deformed Dirac operators on the partitioned manifold
Invoked by the approach based on deformed Dirac operators described in the abstract.

invented entities (1)

vertical Â-cowaist no independent evidence
purpose: Codimension-one invariant measuring a width-like quantity for partitioned manifolds
Newly introduced to extend the infinite vertical Â-cowaist from bands to general cases.

pith-pipeline@v0.9.0 · 5646 in / 1274 out tokens · 42518 ms · 2026-05-20T00:17:11.656308+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean, IndisputableMonolith/Cost/FunctionalEquation.lean reality_from_one_distinction, washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce the vertical Â-cowaist... establish a sharp inequality relating the scalar curvature, the bottom spectrum of the Laplacian, and this invariant... Our approach is based on deformed Dirac operators.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A. ... inf scalgM + 4m/(m-1) λ1(M,gM) ≤ 2m(m-1)/Â-vcw²(M)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 1 internal anchor

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Mingliang Cai, Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set , Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 2, 371–377. MR1071028

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E. Calabi, On manifolds with nonnegative Ricci curvature II , Notices of the American Mathematical Society 22 (1975), no. A, 205

work page 1975
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Simone Cecchini, Callias-type operators in C ∗-algebras and positive scalar curvature on noncompact manifolds, J. Topol. Anal. 12 (2020), no. 4, 897–939. MR4146567

work page 2020
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Cecchini, D

S. Cecchini, D. Räde, and R. Zeidler, Nonnegative scalar curvature on manifolds with at least two ends , J. Topol. 16 (2023), no. 3, 855–876. MR4611408

work page 2023
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Cecchini and R

S. Cecchini and R. Zeidler, Scalar and mean curvature comparison via the Dirac operator , Geom. Topol. 28 (2024), no. 3, 1167–1212. MR4746412

work page 2024
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Differential Geometry 6 (1971/72), 119–128

Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature , J. Differential Geometry 6 (1971/72), 119–128. MR0303460

work page 1971
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Shiu Yuen Cheng, Eigenvalue comparison theorems and its geometric applications , Math. Z. 143 (1975), no. 3, 289–297. MR0378001

work page 1975
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Christopher B. Croke and Bruce Kleiner, A warped product splitting theorem , Duke Math. J. 67 (1992), no. 3, 571–574. MR1181314

work page 1992
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work page arXiv 2025
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Matthew P. Gaffney, A special Stokes’s theorem for complete Riemannian manifolds , Ann. of Math. (2) 60 (1954), 140–145. MR0062490

work page 1954
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3, 489–507

Ezra Getzler, The odd Chern character in cyclic homology and spectral flow , Topology 32 (1993), no. 3, 489–507. MR1231957

work page 1993
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Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures , Functional analysis on the eve of the 21st century, Vol

M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures , Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993), Progr. Math., vol. 132, Birkhäuser Boston, Boston, MA, 1996, pp. 1–213. MR1389019

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Mikhael Gromov and H. Blaine Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds , Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196 (1984). MR0720933

work page 1983
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Sven Hirsch, Demetre Kazaras, Marcus Khuri, and Yiyue Zhang, Spectral torical band inequalities and generalizations of the Schoen-Yau black hole existence theorem , Int. Math. Res. Not. IMRN 4 (2024), 3139–3175. MR4707281

work page 2024
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Ryosuke Ichida, Riemannian manifolds with compact boundary , Yokohama Math. J. 29 (1981), no. 2, 169–177. MR0649619

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Atsushi Kasue, Ricci curvature, geodesics and some geometric properties of Riemannian manifolds with boundary, J. Math. Soc. Japan 35 (1983), no. 1, 117–131. MR0679079

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Kazdan, Deformation to positive scalar curvature on complete manifolds , Math

Jerry L. Kazdan, Deformation to positive scalar curvature on complete manifolds , Math. Ann. 261 (1982), no. 2, 227–234. MR0675736

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Blaine Lawson Jr

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry , Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR1031992

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Math., vol

Matthias Lesch, The uniqueness of the spectral flow on spaces of unbounded self-adjoint Fredholm operators, Spectral geometry of manifolds with boundary and decomposition of manifolds, Contemp. Math., vol. 366, Amer. Math. Soc., Providence, RI, 2005, pp. 193–224. MR2114489

work page 2005
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Peter Li and Luen-Fai Tam, Positive harmonic functions on complete manifolds with nonnegative cur- vature outside a compact set , Ann. of Math. (2) 125 (1987), no. 1, 171–207. MR0873381

work page 1987
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André Lichnerowicz, Spineurs harmoniques , C. R. Acad. Sci. Paris 257 (1963), 7–9 (French). MR0156292

work page 1963
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Daoqiang Liu, Bottom spectrum and Llarull’s theorem on complete noncompact manifolds (2026), avail- able at https://arxiv.org/abs/2601.15043

work page arXiv 2026
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Reine Angew

John Lott, Some obstructions to positive scalar curvature on a noncompact manifold , J. Reine Angew. Math. 829 (2025), 247–283. MR4999591 BOTTOM SPECTRUM, VERTICAL bA-COW AIST AND SCALAR CUR V ATURE RIGIDITY 21

work page 2025
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H. P. McKean, An upper bound to the spectrum of ∆ on a manifold of negative curvature , J. Differential Geometry 4 (1970), 359–366. MR0266100

work page 1970
[32]

Ovidiu Munteanu and Jiaping Wang, Comparison theorems for 3D manifolds with scalar curvature bound, Int. Math. Res. Not. IMRN 3 (2023), 2215–2242. MR4565611

work page 2023
[33]

, Bottom spectrum of three-dimensional manifolds with scalar curvature lower bound , J. Funct. Anal. 287 (2024), no. 2, Paper No. 110457, 41. MR4736650

work page 2024
[34]

Johan Råde, Callias’ index theorem, elliptic boundary conditions, and cutting and gluing , Comm. Math. Phys. 161 (1994), no. 1, 51–61. MR1266069

work page 1994
[35]

Jonathan Rosenberg and Stephan Stolz, Manifolds of positive scalar curvature , Algebraic topology and its applications, Math. Sci. Res. Inst. Publ., vol. 27, Springer, New York, 1994, pp. 241–267. MR1268192

work page 1994
[36]

Pengshuai Shi, The odd-dimensional long neck problem via spectral flow , Int. Math. Res. Not. IMRN 17 (2025), Paper No. rnaf262, 19. MR4951381

work page 2025
[37]

, Spectral flow of Callias operators, odd K-cowaist, and positive scalar curvature , Adv. Math. 479 (2025), Paper No. 110429, 41. MR4929482

work page 2025
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Michael E. Taylor, Partial differential equations I. Basic theory , 2nd ed., Applied Mathematical Sciences, vol. 115, Springer, New York, 2011. MR2744150

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J. Zhu, Calabi-Yau type theorem for complete manifolds with nonnegative scalar curvature (2024), avail- able at https://arxiv.org/abs/2402.15118. Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, China Email address : dqliu@nankai.edu.cn URL: www.dqliu.cn

work page arXiv 2024

[1] [1]

Anghel, On the index of Callias-type operators , Geom

N. Anghel, On the index of Callias-type operators , Geom. Funct. Anal. 3 (1993), no. 5, 431–438. MR1233861

work page 1993

[2] [2]

M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds , Bull. Amer. Math. Soc. 69 (1963), 422–433. MR0157392

work page 1963

[3] [3]

Bär, Metrics with harmonic spinors , Geom

C. Bär, Metrics with harmonic spinors , Geom. Funct. Anal. 6 (1996), no. 6, 899–942. MR1421872

work page 1996

[4] [4]

Curvature-free effects from volume growth and ends-counting and their applications

Y. Bi and J. Zhu, Curvature-free effects from volume growth and ends-counting and their applications (2026), available at https://arxiv.org/abs/2605.12403

work page internal anchor Pith review Pith/arXiv arXiv 2026

[5] [5]

Bernhelm Booss-Bavnbek, Matthias Lesch, and John Phillips, Unbounded Fredholm operators and spec- tral flow , Canad. J. Math. 57 (2005), no. 2, 225–250. MR2124916 20 DAOQIANG LIU

work page 2005

[6] [6]

MR3410545

Jean-Pierre Bourguignon, Oussama Hijazi, Jean-Louis Milhorat, Andrei Moroianu, and Sergiu Moroianu, A spinorial approach to Riemannian and conformal geometry , EMS Monographs in Mathematics, Eu- ropean Mathematical Society (EMS), Zürich, 2015. MR3410545

work page 2015

[7] [7]

Mingliang Cai, Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set , Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 2, 371–377. MR1071028

work page 1991

[8] [8]

Calabi, On manifolds with nonnegative Ricci curvature II , Notices of the American Mathematical Society 22 (1975), no

E. Calabi, On manifolds with nonnegative Ricci curvature II , Notices of the American Mathematical Society 22 (1975), no. A, 205

work page 1975

[9] [9]

Simone Cecchini, Callias-type operators in C ∗-algebras and positive scalar curvature on noncompact manifolds, J. Topol. Anal. 12 (2020), no. 4, 897–939. MR4146567

work page 2020

[10] [10]

Cecchini, D

S. Cecchini, D. Räde, and R. Zeidler, Nonnegative scalar curvature on manifolds with at least two ends , J. Topol. 16 (2023), no. 3, 855–876. MR4611408

work page 2023

[11] [11]

Cecchini and R

S. Cecchini and R. Zeidler, Scalar and mean curvature comparison via the Dirac operator , Geom. Topol. 28 (2024), no. 3, 1167–1212. MR4746412

work page 2024

[12] [12]

Differential Geometry 6 (1971/72), 119–128

Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature , J. Differential Geometry 6 (1971/72), 119–128. MR0303460

work page 1971

[13] [13]

Shiu Yuen Cheng, Eigenvalue comparison theorems and its geometric applications , Math. Z. 143 (1975), no. 3, 289–297. MR0378001

work page 1975

[14] [14]

Croke and Bruce Kleiner, A warped product splitting theorem , Duke Math

Christopher B. Croke and Bruce Kleiner, A warped product splitting theorem , Duke Math. J. 67 (1992), no. 3, 571–574. MR1181314

work page 1992

[15] [15]

Tiarlos Cruz and Almir Silva Santos, Curvature deformations on complete manifolds with boundary (2025), available at https://arxiv.org/abs/2501.10855

work page arXiv 2025

[16] [16]

Gaffney, A special Stokes’s theorem for complete Riemannian manifolds , Ann

Matthew P. Gaffney, A special Stokes’s theorem for complete Riemannian manifolds , Ann. of Math. (2) 60 (1954), 140–145. MR0062490

work page 1954

[17] [17]

3, 489–507

Ezra Getzler, The odd Chern character in cyclic homology and spectral flow , Topology 32 (1993), no. 3, 489–507. MR1231957

work page 1993

[18] [18]

Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures , Functional analysis on the eve of the 21st century, Vol

M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures , Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993), Progr. Math., vol. 132, Birkhäuser Boston, Boston, MA, 1996, pp. 1–213. MR1389019

work page 1993

[19] [19]

, Four lectures on scalar curvature , Perspectives in scalar curvature. Vol. 1, World Sci. Publ., Hackensack, NJ, [2023] ©2023, pp. 1–514. MR4577903

work page 2023

[20] [20]

Blaine Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds , Inst

Mikhael Gromov and H. Blaine Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds , Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196 (1984). MR0720933

work page 1983

[21] [21]

Sven Hirsch, Demetre Kazaras, Marcus Khuri, and Yiyue Zhang, Spectral torical band inequalities and generalizations of the Schoen-Yau black hole existence theorem , Int. Math. Res. Not. IMRN 4 (2024), 3139–3175. MR4707281

work page 2024

[22] [22]

Ryosuke Ichida, Riemannian manifolds with compact boundary , Yokohama Math. J. 29 (1981), no. 2, 169–177. MR0649619

work page 1981

[23] [23]

Atsushi Kasue, Ricci curvature, geodesics and some geometric properties of Riemannian manifolds with boundary, J. Math. Soc. Japan 35 (1983), no. 1, 117–131. MR0679079

work page 1983

[24] [24]

Kazdan, Deformation to positive scalar curvature on complete manifolds , Math

Jerry L. Kazdan, Deformation to positive scalar curvature on complete manifolds , Math. Ann. 261 (1982), no. 2, 227–234. MR0675736

work page 1982

[25] [25]

Blaine Lawson Jr

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry , Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR1031992

work page 1989

[26] [26]

Math., vol

Matthias Lesch, The uniqueness of the spectral flow on spaces of unbounded self-adjoint Fredholm operators, Spectral geometry of manifolds with boundary and decomposition of manifolds, Contemp. Math., vol. 366, Amer. Math. Soc., Providence, RI, 2005, pp. 193–224. MR2114489

work page 2005

[27] [27]

Peter Li and Luen-Fai Tam, Positive harmonic functions on complete manifolds with nonnegative cur- vature outside a compact set , Ann. of Math. (2) 125 (1987), no. 1, 171–207. MR0873381

work page 1987

[28] [28]

André Lichnerowicz, Spineurs harmoniques , C. R. Acad. Sci. Paris 257 (1963), 7–9 (French). MR0156292

work page 1963

[29] [29]

Daoqiang Liu, Bottom spectrum and Llarull’s theorem on complete noncompact manifolds (2026), avail- able at https://arxiv.org/abs/2601.15043

work page arXiv 2026

[30] [30]

Reine Angew

John Lott, Some obstructions to positive scalar curvature on a noncompact manifold , J. Reine Angew. Math. 829 (2025), 247–283. MR4999591 BOTTOM SPECTRUM, VERTICAL bA-COW AIST AND SCALAR CUR V ATURE RIGIDITY 21

work page 2025

[31] [31]

H. P. McKean, An upper bound to the spectrum of ∆ on a manifold of negative curvature , J. Differential Geometry 4 (1970), 359–366. MR0266100

work page 1970

[32] [32]

Ovidiu Munteanu and Jiaping Wang, Comparison theorems for 3D manifolds with scalar curvature bound, Int. Math. Res. Not. IMRN 3 (2023), 2215–2242. MR4565611

work page 2023

[33] [33]

, Bottom spectrum of three-dimensional manifolds with scalar curvature lower bound , J. Funct. Anal. 287 (2024), no. 2, Paper No. 110457, 41. MR4736650

work page 2024

[34] [34]

Johan Råde, Callias’ index theorem, elliptic boundary conditions, and cutting and gluing , Comm. Math. Phys. 161 (1994), no. 1, 51–61. MR1266069

work page 1994

[35] [35]

Jonathan Rosenberg and Stephan Stolz, Manifolds of positive scalar curvature , Algebraic topology and its applications, Math. Sci. Res. Inst. Publ., vol. 27, Springer, New York, 1994, pp. 241–267. MR1268192

work page 1994

[36] [36]

Pengshuai Shi, The odd-dimensional long neck problem via spectral flow , Int. Math. Res. Not. IMRN 17 (2025), Paper No. rnaf262, 19. MR4951381

work page 2025

[37] [37]

, Spectral flow of Callias operators, odd K-cowaist, and positive scalar curvature , Adv. Math. 479 (2025), Paper No. 110429, 41. MR4929482

work page 2025

[38] [38]

Taylor, Partial differential equations I

Michael E. Taylor, Partial differential equations I. Basic theory , 2nd ed., Applied Mathematical Sciences, vol. 115, Springer, New York, 2011. MR2744150

work page 2011

[39] [39]

Shing Tung Yau, Some function-theoretic properties of complete Riemannian manifold and their appli- cations to geometry , Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. MR0417452

work page 1976

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