Newtheorem and theoremstyle test

Michael Downes

December 9, 1999

1 Test of standard theorem styles

Ahlfors’ Lemma gives the principal criterion for obtaining lower bounds on the Kobayashi metric.

Ahlfors’ Lemma. Let ds^{2} = h(z)|dz|^{2} be a Hermitian pseudo-metric on D_{r}, h ∈ C^{2}(D_{r}), with ω the associated (1, 1)-form. If Ric ω ≥ ω on D_{r}, then ω ≤ ω_{r} on all of D_{r} (or equivalently, ds^{2} ≤ ds_{r}^{2}).

Lemma 1.1 (negatively curved families). Let {ds_{1}^{2}, …, ds_{k}^{2}} be a negatively curved family of metrics on D_{r}, with associated forms ω^{1}, …, ω^{k}. Then ω^{i} ≤ ω_{r} for all i.

Then our main theorem:

Theorem 1.2. Let d_{max} and d_{min} be the maximum, resp. minimum distance between any two adjacent vertices of a quadrilateral Q. Let σ be the diagonal pigspan of a pig P with four legs. Then P is capable of standing on the corners of Q iff

σ ≥ \sqrt{d_{max}^{2} + d_{min}^{2}}.  (1)

Corollary 1.3. Admitting reflection and rotation, a three-legged pig P is capable of standing on the corners of a triangle T iff (1) holds.

Remark. As two-legged pigs generally fall over, the case of a polygon of order 2 is uninteresting.

Exercise 1: Generalize Theorem 1.2 to three and four dimensions.

Note 1: This is a test of the custom theorem style ‘note’. It is supposed to have variant fonts and other differences.

B-Theorem 1. Test of the ‘linebreak’ style of theorem heading.

This is a test of a citing theorem to cite a theorem from some other source. Theorem 3.6 in [1]. No hyperlinking available here yet … but that’s not a bad idea for the future.

Proof. Here is a test of the proof environment.

Proof of Theorem 1.2. And another test.

Proof (necessity). And another.

Proof (sufficiency). And another.

2 Test of number-swapping

This is a repeat of the first section but with numbers in theorem heads swapped to the left.

Ahlfors’ Lemma gives the principal criterion for obtaining lower bounds on the Kobayashi metric.

Ahlfors’ Lemma. Let ds^{2} = h(z)|dz|^{2} be a Hermitian pseudo-metric on D_{r}, h ∈ C^{2}(D_{r}), with ω the associated (1, 1)-form. If Ric ω ≥ ω on D_{r}, then ω ≤ ω_{r} on all of D_{r} (or equivalently, ds^{2} ≤ ds_{r}^{2}).

2.1 Lemma (negatively curved families). Let {ds_{1}^{2}, …, ds_{k}^{2}} be a negatively curved family of metrics on D_{r}, with associated forms ω^{1}, …, ω^{k}. Then ω^{i} ≤ ω_{r} for all i.

Then our main theorem:

2.1 Theorem. Let d_{max} and d_{min} be the maximum, resp. minimum distance between any two adjacent vertices of a quadrilateral Q. Let σ be the diagonal pigspan of a pig P with four legs. Then P is capable of standing on the corners of Q iff

σ ≥ \sqrt{d_{max}^{2} + d_{min}^{2}}.  (2)

2.2 Corollary. Admitting reflection and rotation, a three-legged pig P is capable of standing on the corners of a triangle T iff (2) holds.

References

[1] Dummy entry.