• Nenhum resultado encontrado

Rigidez da esfera no espaço euclidiano

N/A
N/A
Protected

Academic year: 2018

Share "Rigidez da esfera no espaço euclidiano"

Copied!
42
0
0

Texto

(1)

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪

❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙

❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼

▼❆❚❊▼➪❚■❈❆

◆❊■▲❍❆ ▼❆❘❈■❆ P■◆❍❊■❘❖

❘■●■❉❊❩ ❉❆ ❊❙❋❊❘❆ ◆❖ ❊❙P❆➬❖ ❊❯❈▲■❉■❆◆❖

(2)

◆❊■▲❍❆ ▼❆❘❈■❆ P■◆❍❊■❘❖

❘■●■❉❊❩ ❉❆ ❊❙❋❊❘❆ ◆❖ ❊❙P❆➬❖ ❊❯❈▲■❉■❆◆❖

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ●❡♦♠❡tr✐❛✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ●❡r✈ás✐♦ ❈♦❧❛r❡s✳

❈♦♦r❞❡♥❛❞♦r✿ Pr♦❢✳ ❉r✳ ❊❞✉❛r❞♦ ✈❛s❝♦♥❝❡❧♦s ❖❧✐✈❡✐r❛ ❚❡✐①❡✐r❛✳

(3)

P✐♥❤❡✐r♦✱ ◆✳ ▼❛r❝✐❛

❳❳❳❳❳ ❘✐❣✐❞❡③ ❞❛ ❊s❢❡r❛ ♥♦ ❊s♣❛ç♦ ❊✉❝❧✐❞✐❛♥♦ ◆❡✐❧❤❛ ▼❛r❝✐❛ P✐♥❤❡✐r♦✳ ✲ ✷✵✶✸✳

✹✶ ❢✳

❉✐ss❡rt❛çã♦ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s✱ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ❋♦rt❛❧❡③❛✱ ✷✵✶✸✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ●❡♦♠❡tr✐❛ ✳

❖r✐❡♥t❛çã♦✿ Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ●❡r✈ás✐♦ ❈♦❧❛r❡s✳

✶✳ ❘✐❣✐❞❡③ ❞❛ ❊s❢❡r❛✳ ✷✳ ❚❡♥s♦r ❞❡ ❯♠❜✐❧✐❝✐❞❛❞❡✳ ✸✳ ❈✉r✈❛t✉r❛s ▼é❞✐❛s ❞❡ ❖r❞❡♥s ❙✉♣❡r✐♦r❡s✳

(4)

◆❡✐❧❤❛ ▼❛r❝✐❛ P✐♥❤❡✐r♦

❘✐❣✐❞❡③ ❞❛ ❡s❢❡r❛ ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ➪r❡❛ ❞❡ ❝♦♥❝❡♥tr❛çã♦✿ ●❡♦♠❡tr✐❛✳

❆♣r♦✈❛❞♦ ❡♠✿ ✴ ✴ ✳

❇❆◆❈❆ ❊❳❆▼■◆❆❉❖❘❆

Pr♦❢✳ ❉r✳ ❆♥tô♥✐♦ ●❡r✈ás✐♦ ❈♦❧❛r❡s ✭❖r✐❡♥t❛❞♦r✮ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ✭❯❋❈✮

Pr♦❢✳ ❉r✳ ●r❡❣ór✐♦ P❛❝❡❧❧✐✲❋❡✐t♦s❛ ❇❡ss❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞♦ ❈❡❛rá ✭❯❋❈✮

(5)
(6)

❆●❘❆❉❊❈■▼❊◆❚❖

❆ ❉❡✉s✱ ♣♦✐s t✉❞♦ ♦ q✉❡ ❛❝♦♥t❡❝❡ ❡♠ ♠✐♥❤❛ ✈✐❞❛ é ❞❡✈✐❞♦ ❛ ❊❧❡✳ ❙❡♠♣r❡ t✐✈❡ ❞✐✜❝✉❧❞❛❞❡s✱ ♠❛s ❣r❛ç❛s ❛ ❉❡✉s t✐✈❡ ❢♦rç❛ ♣❛r❛ ❧✉t❛r ❡ ❝♦♥t♦r♥❛r✲❧❛s✳

❆♦s ♠❡✉s ♣❛✐s✱ ❏♦sé ❍❡♥r✐q✉❡ P✐♥❤❡✐r♦ ❡ ■❧â♥✐❛ ▼❛r✐❛ P✐♥❤❡✐r♦✱ ❛s ♠✐♥❤❛s ✐r♠ãs ❱❡r❛✱ ▲✐❞✐❛♥❡✱ ❘❛❢❛❡❧❛✱ ▲❛r❛ ❡ ❛ ♠❡✉ ✐r♠ã♦ ❑❛②♦ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦✳

➚ ♠✐♥❤❛ ♣r✐♠❛✱ ❆♥tô♥✐❛ ❏♦❝✐✈â♥✐❛ P✐♥❤❡✐r♦✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❡ ♣❡❧♦ ❡stí♠✉❧♦✳ ❆❞❡♠❛✐s✱ ❢♦✐ ✉♠ ❡①❡♠♣❧♦ ❞❡ s✉♣❡r❛çã♦ ❡ ❢♦rç❛ ❞❡ ✈♦♥t❛❞❡ ♥❛ ❜✉s❝❛ ❞❡ ❛❧❝❛ç❛r s❡✉s ♦❜❥❡t✐✈♦s✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❆♥tô♥✐♦ ●❡r✈ás✐♦ ❈♦❧❛r❡s✱ ♣♦r t♦❞❛ ❛ ♣❛❝✐ê♥❝✐❛ q✉❡ t❡✈❡ ❝♦♠✐❣♦ ❡ ♣❡❧♦ ❛♣♦✐♦ q✉❡ ♠❡ ❞❡✉ ❛♦ ❧♦♥❣♦ ❞❡ss❛ ❥♦r♥❛❞❛✳

❆♦s Pr♦❢❡ss♦r❡s ❆❢♦♥s♦✱ ❆❧❡①❛♥❞r❡✱ ❋á❜✐♦✱ ❋❡r♥❛♥❞❛✱ ❘♦❜ér✐♦ ❘♦❣ér✐♦✱ ▲✉q✉és✐♦ ❡ ❖t♦♥ ♣♦r ♠❡ ❛❥✉❞❛r❡♠ ❡♠ ❞✐✈❡rs♦s ♠♦♠❡♥t♦s ❞❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳

❆♦s ♣r♦❢❡ss♦r❡s ❞❡ ❣r❛❞✉❛çã♦ ❋❛❜rí❝✐♦✱ ❏♦s❡r❧❛♥✱ ❉❡♥✐s❡✱ ❏♦♥❛t❛♥✱ ▲♦❡st❡r ❡ ❛♦s ❛♠✐❣♦s ❙❤✐r❧❡②✱ ❑✐❛r❛ ❡ ▼✐❝❤❡❧ q✉❡ ❞❡r❛♠ ❣r❛♥❞❡s ❝♦♥tr✐❜✉✐çõ❡s ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♠❛t❡♠át✐❝❛✳

❆♦s ❝♦❧❡❣❛s ❞♦ ♠❡str❛❞♦✱ ❆♥❞❡rs♦♥✱ ❇r❡♥♦✱ ❏♦sé ❊❞✉❛r❞♦✱ ❘♦❣❡r✱ ●✐❧s♦♥✱ ❉✐❡❣♦ ❊❧♦✐✱ ❲❛♥❞❡r❧❡②✱ ❙❡❧❡♥❡✱ ❘✉✐ ❇r❛s✐❧❡✐r♦✱ ❏♦ã♦ ◆✉♥❡s ✱ ❨✉r❡ ❞♦ ❙❛♥t♦s✱ ❏♦ã♦ ❱✐❝t♦r✱ ◆í❝✉❧❛s✱ ▼❛r❧♦♥✱ ❍❡♥r✐q✉❡ ❇❧❛♥❝♦ ❡ ❊❞s♦ ●❛♠❛ ❡✱ ❛♦s ❝♦❧❡❣❛s ❞♦ ❞♦✉t♦r❛❞♦✱ ❆❞r✐❛♥♦✱ ❘♦♥❞✐♥❡❧❧❡✱ ❈❧❡✐t♦♥✱ ❘❛❢❛❡❧ ❉✐♦❣❡♥❡s✱ ❘❡♥✐✈❛❧❞♦✱ ❆ss✐s✱ ❊❞s♦♥ ❙❛♠♣❛✐♦✱ ❋❛❜✐❛♥❛ ❡ ❏♦ã♦ ❱✐t♦r ♣❡❧❛ ❛♠✐③❛❞❡ ❡ tr♦❝❛ ❞❡ ❡①♣❡r✐ê♥❝✐❛s✳ ◗✉❡ ❉❡✉s ❛❜❡♥ç♦❡ ❛ t♦❞♦s✦

❆♦s ❝♦❧❡❣❛s ❞❛ ❜❛✐❛ ❘♦❞r✐❣♦ ▼❛tt♦s✱ ■t❛♠❛r✱ ❘❡♥❛♥ ❞♦s ❙❛♥t♦s✱ ❘❡♥❛♥ ❇r❛③✱ ❏♦ã♦ ▲✉✐③✱ ❆❞❡♥✐❧s♦♥ ❆r❝❛♥❥♦✱ ❍✉❞s♦♥ ▲✐♠❛ ❡ ❘❛❢❛❡❧ ❆❧✈❡s✱ ♣❡❧♦s ❜♦♥s ♠♦♠❡♥t♦s ✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ♣❡❧❛ ❡❧✉❝✐❞❛çã♦ ❞❡ ❡✈❡♥t✉❛✐s ❞ú✈✐❞❛s✳

➚ ❆♥❞r❡✐❛✱ ●és✐❝❛✱ ❊❧✐③❡✉❞❛✱ ❏ú♥✐♦r✱ ❊r✐✈❛♥ ✱ ❋❡r♥❛♥❞❛ ❡ ❘♦s✐❧❞❛ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ q✉❡ t✐✈❡r❛♠ ❝♦♠✐❣♦✳

(7)

❘❊❙❯▼❖

◆❡st❡ tr❛❜❛❧❤♦✱ ♣r♦✈❛♠♦s ♥♦✈♦s r❡s✉t❛❞♦s ❞❡ r✐❣✐❞❡③ ♣❛r❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲ ❊✐♥st❡✐♥s ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✱ ❜❛s❡❛❞♦✲s❡ ♥♦s r❡s✉❧t❛❞♦s ♣✐♥❝❤✐♥❣ ❞♦ ❛✉t♦✈❛❧♦r✳ ❊♥tã♦✱ ♥ós ❞❡❞✉③✐♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❛♥á❧♦❣♦s ♣❛r❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲ ✉♠❜í❧✐❝❛s ❡ ✉♠❛ ♥♦✈❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ❡s❢❡r❛s ❣❡♦❞és✐❝❛s✳

(8)

❆❇❙❚❘❆❈❚

■♥ t❤✐s ✇♦r❦✱ ✇❡ ♣r♦✈❡ ♥❡✇ r✐❣✐❞✐t② r❡s✉❧ts ❢♦r ❛❧♠♦st✲❊✐♥st❡✐♥ ❤②♣❡rs✉r❢❛❝❡s ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡✱ ❜❛s❡❞ ♦♥ ♣r❡✈✐♦✉s ❡✐❣❡♥✈❛❧✉❡ ♣✐♥❝❤✐♥❣ r❡s✉❧ts✳ ❚❤❡♥✱ ✇❡ ❞❡❞✉❝❡ s♦♠❡ ❝♦♠♣❛r❛❜❧❡ r❡s✉❧t ❢♦r ❛❧♠♦st✲✉♠❜✐❧✐❝ ❤②♣❡rs✉r❢❛❝❡s ❛♥❞ ♥❡✇ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ ❣❡♦❞❡s✐❝ s♣❤❡r❡s✳

(9)

❙❯▼➪❘■❖

✶ ■◆❚❘❖❉❯➬➹❖ ✶

✷ P❘❊▲■▼■◆❆❘❊❙ ✷

✸ ❋❆❚❖❙ ❇➪❙■❈❖❙ ✶✵

✹ ❚❊❖❘❊▼❆❙ ❉❊ ❘■●■❉❊❩ ✷✶

✹✳✶ ❍✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲❊✐♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✷ ❍✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲❯♠❜í❧✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✹✳✸ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

(10)

❈❛♣ít✉❧♦ ✶

■◆❚❘❖❉❯➬➹❖

❙❛❜❡♠♦s ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❆❧❡①❛♥❞r♦✈ q✉❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ♠❡r❣✉❧❤❛❞❛s ❡♠

Rn+1❝♦♠ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥st❛♥t❡ sã♦ ❡s❢❡r❛s ❣❡♦❞és✐❝❛s✳ P♦ré♠✱ ❡st❡ r❡s✉❧t❛❞♦

♥ã♦ é s❡♠♣r❡ ✈❡r❞❛❞❡ ♣❛r❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ✐♠❡rs❛s✱ ✉♠ ❡①❡♠♣❧♦ é ♦ t♦r♦ ❞❡ ❲❡♥t❡✱ ♦ q✉❛❧ é ✉♠ ❡①❡♠♣❧♦ ❞❡ s✉r♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ ❝♦♠ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥st❛♥t❡ ❡♠

R3✱ ♠❛s ♥ã♦ é ✉♠❛ ❡s❢❡r❛ ❣❡♦❞és✐❝❛✳ P❛r❛ ❡st❡ r❡s✉❧t❛❞♦ ✈❛❧❡r ♣❛r❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡s

✐♠❡rs❛s ❞❡ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥st❛♥t❡ ✉♠❛ ❤✐♣ót❡s❡ ❛❞❝✐♦♥❛❧ é ♥❡❝❡ssár✐❛✳ ❯♠❛ ❝♦♥❞✐çã♦ é ❞❛❞❛ ♣❡❧♦ t❡♦r❡♠❛ ❞❡ ❍♦♣❢ ❬✼❪✱ ❛ q✉❛❧ ❞✐③ q✉❡ ❡s❢❡r❛s ✐♠❡rs❛s ❝♦♠ ❝✉r✈❛t✉r❛s ♠é❞✐❛s ❝♦♥st❛♥t❡s ❡♠ Rn+1 sã♦ ❡s❢❡r❛s ❣❡♦❞és✐❝❛s✳

◆❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ ♥♦✈♦ t❡♦r❡♠❛ ❞❡ r✐❣✐❞❡③ ♣❛r❛ ❡s❢❡r❛s✱ ❞❡♠♦♥str❛❞♦ ♣♦r ❏✉❧✐❛♥ ❘♦t❤ ❬✶✷❪✱ ♦♥❞❡ é s✉❜st✐t✉✐❞❛ ❛ s✉♣♦s✐çã♦ t♦♣ó❧♦❣✐❝❛ ✭♠❡r❣✉❧❤♦✮ ♣❡❧❛ s✉♣♦s✐çã♦ ♠étr✐❝❛ ✭❝✉r✈❛t✉r❛ ❡s❝❛❧❛r✮✳ Pr❡❝✐s❛♠❡♥t❡✱ é ❢á❝✐❧ ✈❡r q✉❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❞❡ Rn+1 ❝♦♠♣❛❝t❛s ❝♦♠ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥st❛♥t❡ ❡

❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ❝♦♥st❛♥t❡ sã♦ ❡s❢❡r❛s ❣❡♦❞és✐❝❛s✳ ❊st❡ r❡s✉❧t❛❞♦ ✈❡♠ ❞♦ ❢❛t♦ q✉❡ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❞❡ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥st❛♥t❡ ❡ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ❝♦♥st❛♥t❡ sã♦ t♦t❛❧♠❡♥t❡ ✉♠❜í❧✐❝❛s✳ ❆q✉✐✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ ♥♦✈♦ r❡s✉❧t❛❞♦ ❞❡ r✐❣✐❞❡③ ❝♦♠ ✉♠❛ ❤✐♣ót❡s❡ ♠❛✐s ❢r❛❝❛ s♦❜r❡ ❛ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r✳ ▼♦str❛r❡♠♦s✿

❚❡♦r❡♠❛ ✶✳✶ ✭❬✶✷❪✱ ❚❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧✮

❙❡❥❛ (Mn, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ r✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛✱ ♦r✐❡♥t❛❞❛ s❡♠ ❜♦r❞♦ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ✐♠❡rs❛ ❡♠Rn+1✳ ❙❡❥❛h >0θ]0,1[✳ ❊♥tã♦ ❡①✐st❡ ǫ(n, h, θ)>0

❞❡ ♠♦❞♦ q✉❡ s❡✿ ✶✳ H =h

✷✳ |Scals| ≤ǫ,

♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ s✱ ❡♥tã♦ M é ❛ ❡s❢❡r❛ Sn 1

h

❝♦♠ s✉❛ ♠étr✐❝❛ ♣❛❞rã♦✳ Pr♦✈❛r❡♠♦s t❛♠❜é♠ ✉♠ ♥♦✈♦ t❡♦r❡♠❛ ❞❡ r✐❣✐❞❡③ ♣❛r❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲ ❊✐♥st❡✐♥✱ ❞❡❞✉③✐r❡♠♦s ❞❡st❡ t❡♦r❡♠❛ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ♣❛r❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲✉♠❜í❧✐❝❛s✱ ♦❜t❡r❡♠♦s r❡s✉❧t❛❞♦s ♣❛r❛ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r q✉❛s❡ ❝♦♥st❛♥t❡ ❡ ❝✉r✈❛t✉r❛ ♠é❞✐❛ q✉❛s❡ ❝♦♥st❛♥t❡✱ ❡ ❡♥tã♦ ❝♦♥❝❧✉✐r❡♠♦s ♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧✳

(11)

❈❛♣ít✉❧♦ ✷

P❘❊▲■▼■◆❆❘❊❙

◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❡ ❞❡✜♥✐çõ❡s q✉❡ s❡rã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡st✉❞♦ q✉❡ s❡rá ❢❡✐t♦ ♥♦s ❝❛♣ít✉❧♦s s✉❜s❡q✉❡♥t❡s✳

■♥❞✐❝❛r❡♠♦s ♣♦r X(M)♦ ❝♦♥❥✉♥t♦ ❞♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❞❡ ❝❧❛ss❡ C∞ ❡♠ M

❡ ♣♦rD(M) ♦ ❛♥❡❧ ❞❛s ❢✉♥çõ❡s r❡❛✐s ❞❡ ❝❧❛ss❡ C∞ ❞❡✜♥✐❞❛s ❡♠M

❉❡✜♥✐çã♦ ✷✳✶ ❯♠❛ ❝♦♥❡①ã♦ ❛✜♠ ∇ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡♥❝✐á✈❡❧ M é ✉♠❛ ❛♣❧✐❝❛çã♦

∇:X(M)× X(M)−→ X(M)

q✉❡ s❡ ✐♥❞✐❝❛ ♣♦r (X, Y)−→ ∇∇ XY ❡ q✉❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

(i)∇f X+gYZ =f∇XZ+g∇YZ

(ii)X(Y +Z) =∇XY +∇XZ

(iii)X(f Y) = f∇XY +X(f)Y,

♣❛r❛ ❝❛❞❛ X, Y, Z ∈ X(M) ❡ f, g ∈ D(M)✳

❉❡✜♥✐çã♦ ✷✳✷ ❙❡❥❛ f :M −→Rn+1 ✉♠❛ ✐♠❡rsã♦✳ ❙❡X, Y sã♦ ❝❛♠♣♦s ❧♦❝❛✐s ❡♠

M✱

B(X, Y) = XY − ∇XY é ✉♠ ❝❛♠♣♦ ❧♦❝❛❧ ❡♠ Rn+1 ♥♦r♠❛❧ ❛ M✳

❉❡✜♥✐çã♦ ✷✳✸ ❯♠❛ ❝♦♥❡①ã♦ ❛✜♠ ∇ ❡♠ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞✐❢❡r❡❝✐á✈❡❧ M é ❞✐t❛ s✐♠étr✐❝❛ q✉❛♥❞♦

∇XY − ∇YX = [X, Y]∀X, Y ∈ X(M) ✳

■♥❞✐❝❛r❡♠♦ ♣♦r X(U) ♦ ❝♦♥❥✉♥t♦ ❞♦s ❝❛♠♣♦s ❞❡ ✈❡t♦r❡s ❞❡ ❝❧❛ss❡ C∞ ❡♠ U

❛❜❡rt♦ ❞❡ M ❡ ♣♦r X(U)⊥ ♦s ❝❛♠♣♦s ❞✐❢❡r❡♥❝✐á✈❡✐s ❡♠ U ❞❡ ✈❡t♦r❡s ♥♦r♠❛✐s ❛

f(U)U✳

Pr♦♣♦s✐çã♦ ✷✳✹ ❙❡ X, Y ∈ X(U)✱ ❛ ❛♣❧✐❝❛çã♦ B : X(U)× X(U) −→ X(U)⊥

❞❛❞❛ ♣♦r✿

(12)

❉❡♠♦♥str❛çã♦✿

Pr✐♠❡r♦ ♣r♦✈❛r❡♠♦s q✉❡ B é ❜✐❧✐♥❡❛r✱✐st♦ é✱ ✶✳ B(f X1+gX2, Y1) =f B(X1, Y1) +gB(X2, Y1)

✷✳ B(X1, f Y1+gY2) = f B(X1, Y1) +gB(X1, Y2)

✸✳ B(f X1, Y1) = f B(X1, Y1)

✹✳ B(X1, f Y1) = f B(X1, Y1)

f, g ∈ D(U)✱ X1, X2, Y1, Y2 ∈ X(U).

Pr♦✈❛ ❞❡ (1)

❖❜s❡r✈❡ q✉❡✱

B(f X1+gX2, Y1) = ∇f X1+gX2Y1− ∇f X1+gX2Y1.

P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ (i) ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❝♦♥❡①ã♦ ❛✜♠ ✭❈✳❆✮✱ t❡♠♦s✿

B(f X1+gX2, Y1) =f∇X1Y1+g∇X2Y1−f∇X1Y1−g∇X2Y1

=f B(X1, Y1) +gB(X2, Y1).

P♦r ✉♠ ❝á❧❝✉❧♦ ❛♥á❧♦❣♦✱ t❡♠♦s✿

B(X1, f Y1+gY2) =f B(X1, Y1) +gB(X1, Y2).

❆❣♦r❛✱ ♣r♦✈❛r❡♠♦s (2)✱

B(f X1, Y1) =∇f X1Y1− ∇f X1Y1 =f B(X, Y)

❋✐♥❛❧♠❡♥t❡✱ ♣r♦✈❛r❡♠♦s (3)✳ ■♥❞✐❝❛♥❞♦ ♣♦r f ✉♠❛ ❡①t❡♥sã♦ ❞❡ f ❛ U ❛❜❡rt♦ ❞❡Rn+1✱ t❡r❡♠♦s✿

B(X1, f Y1) = ∇X1(f Y1)− ∇X1(f Y1).

P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ (iii) ❞❛ ❞❡✜♥✐çã♦ ✶✳✶ t❡♠♦s

∇X1(f Y1) =f∇X1Y1+X1(f Y1).

❉❛í✱

∇X1(f Y1)− ∇X1(f Y1) = f∇X1Y1−f∇X1Y1+X1(f)Y1−X1(f)Y1.

❈♦♠♦ ❡♠ M✱f =f ❡X1 (f) =X1(f)✱ ❝♦♥❝❧✉✐♠♦s q✉❡ ❛s ❞✉❛s ú❧t✐♠❛s ♣❛❝❡❧❛s

s❡ ❛♥✉❧❛♠✱ ❞♦♥❞❡ B(X1, f(Y1)) = f B(X1, Y1)✳ ▲♦❣♦ B é ❜✐❧✐♥❡❛r✳

❆❣♦r❛✱ ♠♦str❛r❡♠♦s q✉❡ B é s✐♠étr✐❝❛✳ ❯t✐❧✐③❛♥❞♦ ❛ s✐♠❡tr✐❛ ❞❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛✱ ♦❜t❡♠♦s✿

(13)

∇XY = [X, Y] +∇YX

∇XY = [X, Y] +∇YX. ▲♦❣♦✱

B(X, Y) =YX+ [X, Y]− ∇YX−[X, Y].

❈♦♠♦ ❡♠ M✱ [X, Y] = [X, Y]✱ ❝♦♥❝❧✉í♠♦s q✉❡ B(X, Y) =B(Y, X)✳ P♦rt❛♥t♦✱

B é s✐♠étr✐❝❛✳

❉❡✜♥✐çã♦ ✷✳✺ ❙❡❥❛ Sν : TpM −→ TpM ✉♠❛ ❛♣❧✐❝❛çã♦ ❧✐♥❡❛r ❛✉t♦✲❛❞❥✉♥t❛ ❞❛❞❛ ♣♦r✿

hSνX, Yi=hB(x, y), νi.

Pr♦♣♦s✐çã♦ ✷✳✻ ❙❡❥❛ p ∈ M✱ x ∈ TpM ❡ ν ∈ (TpM)⊥✳ ❙❡❥❛ N ✉♠❛ ❡①t❡♥sã♦

❧♦❝❛❧ ❞❡ ν ♥♦r♠❛❧ ❛ M✳ ❊♥tã♦

Sν(x) = −(∇xN)T. ❉❡♠♦♥str❛çã♦✿

❙❡❥❛ y ∈ TνM ❡ X, Y ❡①t❡♥sõ❡s ❧♦❝❛✐s ❞❡ x, y✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ t❛♥❣❡♥t❡ ❛ M✳ ❊♥tã♦ hN, Yi= 0 ❡♠ M✱ ❡ ♣♦rt❛♥t♦

XhY, Ni=h∇XY, Ni+hY,∇XNi= 0. ❈♦♠♦✱ h−(XY)T, Ni= 0✳ ❚❡♠♦s✱

h∇XY −(∇XY)T, Ni=−h∇XN, Yi

❡♠ p∈M hB(x, y), νi=−h∇xN(p), yi✳ ❈♦♠♦ ∇xN(p)♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡

t❛♥❣❡♥t❡ t♦♠❡ (xN(p))T✳ ▲♦❣♦✱

hB(x, y), νi=−h(∇xN(p))T, yi

P♦rt❛♥t♦✱

Sν(x) = −(∇xN)T.

❉❡✜♥✐çã♦ ✷✳✼ ∇⊥ é ❝❤❛♠❛❞❛ ❝♦♥❡①ã♦ ♥♦r♠❛❧ ❞❛ ✐♠❡rsã♦✱♦♥❞❡✿

(14)

❉❡✜♥✐çã♦ ✷✳✽ ❙❡❥❛ (Mn, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ n✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛✱ ♦r✐❡♥t❛❞❛✱ s❡♠ ❜♦r❞♦✱ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ✐♠❡rs❛ ❡♠ ✉♠ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦ (n+ 1)✲❞✐♠❡♥s✐♦♥❛❧✳ ❆ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ B ❞❛ ✐♠❡rsã♦ é ❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r s✐♠étr✐❝❛ ❞❡✜♥✐❞❛ ♣♦r✿

B(Y, Z) =g(Yν, Z)

♦♥❞❡ ∇ é ❛ ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛ s♦❜r❡ Rn+1 ❡ ν é ♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ✉♥✐tár✐♦ ♥♦r♠❛❧ s♦❜r❡ M✳

❆❣♦r❛✱ ♠♦str❛r❡♠♦s q✉❡ B(Y, Z) = g(Yν, Z) é ❜✐❧✐♥❡❛r ❡ s✐♠étr✐❝❛✳ ❖❜s❡r✈❡ q✉❡✱

B(Y, Z, ν) =hB(Y, Z), νi=B(Y, Z) = g(Yν, Z). P♦✐s ♣❡❧❛ ❡q✉❛çã♦ ❞❛ ❞❡✜♥✐çã♦ (2.7)✱ t❡♠♦s✿

∇Yν =∇⊥Yν+ (∇Yν)T. ❆♣❧✐❝❛♥❞♦−g(Yν, Z). ❚❡♠♦s✱

−g(∇Yν, Z) = −g(∇Y⊥ν, Z)−g((∇YV)T, Z)

−g(Yν, Z) = g(−(∇Yν)T, Z). ❈♦♠♦ Sν(Y) = (Yν)T✱ t❡♠♦s✿

−g(∇Yν, Z) =g(Sν(Y), Z) = g(B(Y, Z), ν) =B(Y, Z). ❈♦♠♦ B é ❜✐❧✐♥❡❛r ❡ s✐♠étr✐❝❛✳ ❉❛í✱

B(Y, Z) =g(Yν, Z) é ❜✐❧✐♥❡❛r ❡ s✐♠étr✐❝❛✳

❆ ♣❛rt✐r ❞❛ ❞❡✜♥✐çã♦ ❞❡ B✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛✱

H = 1

ntr(B),

❡ ❞❡ ❢♦r♠❛ ♠❛✐s ❣❡r❛❧ ❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛s ❞❡ ♦r❞❡♠ ♠❛✐♦r✱

Hr = 1n

r

σr(k1, ..., kn).

❖♥❞❡σr é ♦ r✲és✐♠♦ ♣♦❧✐♥ó♠✐♦ s✐♠étr✐❝♦ ❡k1, ..., knsã♦ ❛s ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s

❞❛ ✐♠❡rsã♦✳ ❈♦♥✈❡♥❝✐♦♥❛♠♦s ❛✐♥❞❛ H0 = 1✳ ◆♦t❡ q✉❡ H1 = H ❡ ❞❛ ❡q✉❛çã♦ ❞❡

●❛✉ss

H2 =

1

n(n+ 1)Scal.

(15)

❉❡✜♥✐çã♦ ✷✳✾ ❙❡❥❛ f : Mn −→ Rn+1 ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❡ ki ❝♦♠ i =

1,2, ..., n ❛s ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s ❡♠ ✉♠ ♣♦♥t♦ ❛r❜✐trár✐♦ ❞❡ M✳ ❆ r✲és✐♠❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ Hr ❞❡ f é ❞❡✜♥✐❞❛ ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡✿

Pn(t) = (1 +tk1)...(1 +tkn) = 1 +

n

1

H1t+...+

n n

Hntn. ✭✷✳✶✮ ♣❛r❛ t♦❞♦ t r❡❛❧✳

❖❜s❡r✈❡ q✉❡✱

σr=

n r

Hr.

❉❡✜♥✐çã♦ ✷✳✶✵ ❙❡❥❛ f : Mn −→ Rn+1 ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ✐♠❡rs❛✱ ♦r✐❡♥tá✈❡❧✳

◆ós ❞❡♥♦t❛♠♦s ♣♦r ∆ ♦ ❧❛♣❧❛❝✐❛♥♦ ❞❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛ s♦❜r❡ M✳

Pr♦♣♦s✐çã♦ ✷✳✶✶ ❙❡ f : Mn Rn+1 é ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛✱ ν é ✉♠ ❝❛♠♣♦

♥♦r♠❛❧ ✉♥✐tár✐♦ ❣❧♦❜❛❧♠❡♥t❡ ❞❡✜♥✐❞♦✱ ❡♥tã♦

∇|f|2 = 2f⊤ ❡

|f|2 = 2n(1 +Hhf, νi)

❉❡♠♦♥str❛çã♦✿

❙❡❥❛ e1, ..., en ✉♠ r❡❢❡r❡♥❝✐❛❧ ♠ó✈❡❧ ❡♠ ✉♠ ❛❜❡rt♦ ❞❡ M✳ ◆♦t❡ ♣r✐♠❡✐r♦ q✉❡

∇|f|2 =ekhf, fiek = 2h∇ekf, fiek= 2hek, fiek = 2f

,

♦♥❞❡ f⊤ =f − hf, νiν é ❛ ❝♦♠♣♦♥❡♥t❡ t❛♥❣❡♥t❡ ❞❡ f s♦❜r❡ M é ❝♦♥❡①ã♦

r✐❡♠❛♥♥✐❛♥❛✳ P♦rt❛♥t♦✱ t❡♠♦s ❡♠ pq✉❡

∆|f|2 =h∇

ek(∇|f|

2), eki

=h∇ek(∇|f|

2), ek

i= 2h∇ek(f

), eki

= 2h∇ek(f − hf, νiν), eki

= 2hekekhf, νiν− hf, νi∇ekν, eki

= 2(n+hf, νih−∇ekν, eki)

= 2n(1 +Hhf, νi)

❈♦r♦❧ár✐♦ ✷✳✶✷ ◆❛s ❤í♣♦t❡s❡s ❞❛ ♣r♦♣♦s✐çã♦ (2.11)✱ s❡ M é ❝♦♠♣❛❝t❛ ❡♥tã♦

Z

M

(16)

❉❡♠♦♥str❛çã♦✿

■♥t❡❣r❛♥❞♦ ❛ ú❧t✐♠❛ ❡q✉❛çã♦ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ❡ ✉t✐❧✐③❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❛ ❉✐✈❡r❣ê♥❝✐❛ ♦❜t❡♠♦s

Z

M

(1 +Hhf, νi)dvg = 0.

❉❡✜♥✐çã♦ ✷✳✶✸ ❙❡❥❛ t✉♠ ♥ú♠❡r♦ r❡❛❧ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ♣❛r❛❧❡❧❛ ft é ❞❛❞❛ ♣♦r✿

ft(p) = expf(p)(−tν(p)) =f(p)−tN(p).

❆❣♦r❛✱ s❡ e1, ..., en sã♦ ❞✐r❡çõ❡s ♣r✐♥❝✐♣❛✐s ❡♠ ✉♠ ♣♦♥t♦ ❞❡ M✱ t❡♠♦s✿

(ft)(ei) = (1 +tki)ei;i= 1, ..., n. ✭✷✳✸✮ ❉❡ ✭✷✳✸✮✱ ❝♦♥❝❧✉í♠♦s q✉❡ νé ✉♠ ❝❛♠♣♦ ♥♦r♠❛❧ ✉♥✐tár✐♦ ❞❛ ✐♠❡rsã♦ft✳ ❙❡❥❛Bt ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ft ❝♦♠ r❡s♣❡✐t♦ ❛ ν✱ H(t)s✉❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❡

dvg ❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛ ❡♠M ❞❡ft✳ ❯s❛♥❞♦ ✭✷✳✸✮✱ ✈❡♠✿

dvgt = (1 +tk1)...(1 +tkn)dvg =Pn(t)dvg. ✭✷✳✹✮

❆❧é♠ ❞✐ss♦✱ Bt é ❞❛❞❛ ♣♦r✿

Bt((ft)(v),(ft)(w)) =−hν(v),(ft)(w)i

❈♦♠ v, w t❛♥❣❡♥t❡s ❛ M✳ ❈♦♠♦ e1, ..., en sã♦ ❞✐r❡çõ❡s ♣r✐♥❝✐♣❛✐s ❞❡ ft ❡ s✉❛s

❝♦rr❡s♣♦♥❞❡♥t❡s ❝✉r✈❛t✉r❛s ♣r✐♥❝✐♣❛✐s sã♦ ❞❛❞❛s ♣♦r✿

ki(t) = ki 1 +tki P♦rt❛♥t♦✱ ❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ H(t) ❞❛ ✐♠❡rsã♦ ft é✿

H(t) = 1

n P′

n(t) Pn(t) =

n

1

H1+ 2 n2

H2t+...+n nn

Hntn−1

nPn(t) . ✭✷✳✺✮

❉❡✜♥✐çã♦ ✷✳✶✹ (L1 norma) ❖ ❢✉♥❝✐♦♥❛❧ k.k : L1(R) −→ R ❞❡✜♥✐❞❛ ♣♦r

kfk=R

|f| é ❝❤❛♠❛❞❛ ❛ ♥♦r♠❛ L1(R) ♦✉ ❛ L1 norma✳

❉❡✜♥✐çã♦ ✷✳✶✺ Lp(R)P❛r❛ ✉♠ r❡❛❧p0✱ ♥ós ❞❡♥♦t❛♠♦s ♣♦r Lp(R)♦ ❡s♣❛ç♦ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s f ❞❡ ✈❛❧♦r❡s ❝♦♠♣❧❡①♦s ❧♦❝❛❧♠❡♥t❡ ✐♥t❡❣rá✈❡❧ t❛❧ q✉❡|f|p L1(R)

♦♥❞❡ kfkp = R

|f|p1/p

❚❡♦r❡♠❛ ✷✳✶✻ ❙❡❥❛ 1 p≤ ∞✱ 1 q ≤ ∞✱ ❡ 1

p +

1

q = 1✳ ❙❡ f ∈ L p

R ❡

g ∈Lq(R)✳ ❊♥tã♦ f gL1(R) kf.gk

(17)

❙❡❥❛ f : Mn Mn+m ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛✳ ❚❡♠♦s ❡♠ ❝❛❞❛ p M ❞❡❝♦♠♣♦s✐çã♦

TpM =TpM (TpM)⊥,

q✉❡ ✈❛r✐❛ ❞✐❢❡r❡♥❝✐❛✈❡❧♠❡♥t❡ ❝♦♠ p✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡✱ ❧♦❝❛❧♠❡♥t❡✱ ❛ ♣❛rt❡ ❞♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡T M q✉❡ s❡ ♣r♦❥❡t❛ s♦❜r❡M s❡ ❞❡❝♦♠♣õ❡ ❡♠ ✉♠ ✜❜r❛❞♦ t❛♥❣❡♥t❡ T M ❡ ❡♠ ✉♠ ✜❜r❛❞♦ ♥♦r♠❛❧ T M⊥✳ ❆❞❡♠❛✐s✱ ✉s❛r❡♠♦s

❛s ❧❡tr❛s ❧❛t✐♥❛s X, Y, Z, ❡t❝✳✱ ♣❛r❛ ✐♥❞✐❝❛r ♦s ❝❛♠♣♦s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡ ✈❡t♦r❡s t❛♥❣❡♥t❡s ❡ ❛s ❧❡tr❛s ❣r❡❣❛s ξ, η, ζ ❡t❝✳✱ ♣❛r❛ ✐♥❞✐❝❛r ♦s ❝❛♠♣♦s ❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡ ✈❡t♦r❡s ♥♦r♠❛✐s✳ ❆ ❝♦♠♣♦♥❡♥t❡ ♥♦r♠❛❧ ❞❡ ∇Xη✱ q✉❡ s❡rá ❝❤❛♠❛❞❛ ❛ ❝♦♥❡①ã♦ ♥♦r♠❛❧∇⊥ ❞❛ ✐♠❡rsã♦✳ ❱❡r✐✜❝❛✲s❡ ❢❛❝✐❧♠❡♥t❡ q✉❡ ❛ ❝♦♥❡①ã♦ ♥♦r♠❛❧ ♣♦ss✉✐ ❛s

♣r♦♣r✐❡❞❛❞❡s ✉s✉❛✐s ❞❡ ✉♠❛ ❝♦♥❡①ã♦✳ ◆♦ ✜❜r❛❞♦ t❛♥❣❡♥t❡✱ ✐♥tr♦❞✉③✲s❡ ❛ ♣❛rt✐r ❞❡

∇⊥ ✉♠❛ ♥♦çã♦ ❞❡ ❝✉r✈❛t✉r❛ ♥♦ ✜❜r❛❞♦ ♥♦r♠❛❧ q✉❡ é ❝❤❛♠❛❞❛ ❝✉r✈❛t✉r❛ ♥♦r♠❛❧

R⊥ ❞❛ ✐♠❡rsã♦ ❡ ❞❡✜♥✐❞❛ ♣♦r✿

R⊥(X, Y)η=∇⊥

y∇⊥Xη− ∇⊥X∇Y⊥η+∇⊥[X,Y]η. ❈♦♠ ✐ss♦✱ t❡♠♦s ❛ s❡❣✉✐♥t❡ ♣r❡♣♦s✐çã♦✿

Pr♦♣♦s✐çã♦ ✷✳✶✼ ❉❛❞❛ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ f :Mn Mn+m ❝♦♠ X, Y, Z

X(M)✱ η, ζ ∈ X(M)⊥ ❡ R⊥ ♦ ♦♣❡r❛❞♦r ❞❡ ❝✉r✈❛t✉r❛ ♥♦r♠❛❧ ❞❛ ✐♠❡rsã♦✳ ❊♥tã♦

s❡ ✈❡r✐✜❝❛✿

✭❛✮ ❊q✉❛çã♦ ❞❡ ●❛✉ss

hR(X, Y)Z, Ti=hR(X, Y)Z, Ti − hB(Y, T), B(X, Z)i+hB(X, T), B(Y, Z)i. ✭❜✮ ❊q✉❛çã♦ ❞❡ ❘✐❝❝✐

hR(X, Y)η, ζi − hR⊥(X, Y)η, ζi=h[Sη, Sζ]X, Yi, ♦♥❞❡ [Sη, Sζ] ✐♥❞✐❝❛ ♦ ♦♣❡r❛❞♦r SηSη ✳

❆s ❡q✉❛çõ❡s ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r sã♦ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ❛s ❡q✉❛çõ❡s ❢✉♥❞❛♠❡♥t❛✐s ❞❡ ✉♠❛ ✐♠❡rsã♦ ✐s♦♠❡tr✐❝❛✳

❉❡♥♦t❛♥❞♦✿

B(X, Y, η) =hB(X, Y), ηi.

Pr♦♣♦s✐çã♦ ✷✳✶✽ ✭❊q✉❛çã♦ ❞❡ ❈♦❞❛③③✐✮✳ ❈♦♠ ❛ ♥♦t❛çã♦ ❛❝✐♠❛✱

hR(X, Y), Z, ηi= (YB)(X, Z, η)−(∇XB)(Y, Z, η).

Pr♦♣♦s✐çã♦ ✷✳✶✾ ✭❬✶✼❪✱ Pr♦♣♦s✐çã♦ ✼✳✹✮ ❙✉♣♦♥❤❛ f :M N é ✉♠❛ ✐♠❡rsã♦ ✐♥❥❡t✐✈❛✳ ❙❡ M é ❝♦♠♣❛❝t♦✱ ❡♥tã♦ f é ♠❡r❣✉❧❤♦ ❞✐❢❡r❡♥❝✐á✈❡❧✳

(18)

Pr♦♣♦s✐çã♦ ✷✳✷✶ ❙❡❥❛ f :M −→ Rn+1 ✉♠❛ ✐♠❡rsã♦ ✐s♦♠étr❝❛ ✉♠❜í❧✐❝❛ ❞❡ ✉♠❛

✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♥❡①❛ Mn ❡♠ Rn+1✳ ❊♥tã♦✱ f(M) é ✉♠ s✉❜❝♦♥❥✉♥t♦

❛❜❡rt♦ ❞❡ ✉♠ ❤✐♣❡r♣❧❛♥♦ ❛✜♠ ♦✉ ❞❡ ✉♠❛ ❡s❢❡r❛✳ ❉❡♠♦♥str❛çã♦✿

❊s❝♦❧❤❛ ✉♠ ♣♦♥t♦ x∈M ❡ ✉♠ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ♥♦r♠❛❧ ✉♥✐tár✐♦ ν ❞❡✜♥✐❞♦ ❡♠ ❛❧❣✉♠❛ ✈✐③✐♥❤❛♥ç❛U ❞❡ x✳ ❉❡s❞❡ q✉❡f é ✉♠❜í❧✐❝❛✱ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦λ:U R t❛❧ q✉❡Sν =λI ❡♠ U✱ ♦♥❞❡I é ♦ t❡♥s♦r ✐❞❡♥t✐❞❛❞❡✳ ❡♠ ♣❛rt✐❝✉❧❛r❀ λ= n1tr(Sν)é

❞✐❢❡r❡❝✐á✈❡❧✳ ❉❛❞♦s ♦s ❝❛♠♣♦s ✈❡t♦r✐❛✐sX, Y ∈T U✱ s❡❣✉❡ ❞❛s ❡q✉❛çõ❡s ❞❡ ❈♦❞❛③③✐

q✉❡✱

X(λ)Y =Y(λ)X.

❚♦♠❛♥❞♦X ❡Y ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♥ós ❝♦♥❝❧✉✐♠♦s q✉❡λ é ❝♦♥st❛♥t❡ s♦❜r❡ U✳

❙❡λ= 0✱ ❛ ❢ór♠✉❧❛ ❞❡ ❲❡✐♥❣t❡♥ ♠♦str❛ q✉❡Xν = 0♣❛r❛ t♦❞♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧ X T M|U✳ P♦rt❛♥t♦✱ ν é ❝♦♥st❛♥t❡ ❡♠ Rn+1✳ ❆❣♦r❛✱ ❞❛❞♦ q✉❛❧q✉❡r x U✱ ❡ q✉❛❧q✉❡r ❝✉r✈❛ ❞✐❢❡r❡♥❝✐á✈❡❧ γ : [0,1]→U ❧✐❣❛♥❞♦x àx✱ ♥ós t❡♠♦s✿

d

dth(f oγ)(t), ν(γ(t))i=hdf γ

(t), νi= 0

■st♦ ♠♦str❛ q✉❡ h(f oγ)(t), νi é ❝♦♥st❛♥t❡✳ P♦rt❛♥t♦✱ f(U) ❡stá ❝♦♥t✐❞♦ ♥♦

❤✐♣❡r♣❧❛♥♦ ♣❛ss❛♥❞♦ ♣♦rf(x)❡ ♥♦r♠❛❧ à ν✳ ❙❡ λ6= 0 ❡♠ U✱ ♥ós t❡♠♦s✿

∇X(f +λ−1ν) =df(X)−λ−1Sν(X) =X−X = 0,

♣❛r❛ t♦❞♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧ X ∈ T U✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠ ♣♦♥t♦ c ∈Rn+1 t❛❧ q✉❡

f(y) +λ−1νy = c ♣❛r❛ t♦❞♦ y U✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ f(U) ❡stá ❝♦♥t✐❞♦ ♥❛

❡s❢❡r❛ ❝♦♠ ❝❡♥tr♦c ❡ r❛✐♦ |λ|−1

(19)

❈❛♣ít✉❧♦ ✸

❋❆❚❖❙ ❇➪❙■❈❖❙

◆❡st❡ ❝❛♣ít✉❧♦✱ ♣r♦✈❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞♦s ♥❛s ❞❡♠♦♥str❛çõ❡s ❞♦s t❡♦r❡♠❛s ❞❡ r✐❣✐❞❡③ ❞♦ ❝❛♣ít✉❧♦ ✹✳ ◆❛s ❞❡♠♦♥str❛çõ❡s ❞♦s r❡s✉❧t❛❞♦s ❞❡st❡ ❝❛♣ít✉❧♦✱ ♥ós r❡❝♦r❡r❡♠♦s ❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✳

▲❡♠❛ ✸✳✶ ✭❋ór♠✉❧❛s ❞❡ ▼✐♥❦♦✇s❦✐✮ ❙❡❥❛ f : Mn −→ Rn+1 ✉♠❛

❤✐♣❡rs✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛✱ ♦r✐❡♥tá✈❡❧ ✐♠❡rs❛ ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦✳ ❊♥tã♦✱ ♣❛r❛

1≤r≤n✱

Z

M

(Hr1−Hrh(f, ν)i)dvg = 0.

❉❡♠♦♥str❛çã♦✿

❙❛❜❡♠♦s q✉❡ (2.2) é ✈á❧✐❞❛ ♣❛r❛ q✉❛❧q✉❡r ❤✐♣❡rs✉♣❡r❢í❝✐❡✱ ❡♥tã♦ ♣❛r❛|t|< ǫ

Z

M

(1 +H(t)hft, νi)dvgt = 0.

❖❜s❡r✈❡ q✉❡ ♣♦r ✭✷✳✹✮✱ t❡♠♦s✿

Z

M

(1+H(t)hft, νi)dvgt =

Z

M

(1+H(t)hft, νi)Pn(t)dvg =

Z

M

(Pn(t)+H(t)Pn(t)hft, νi)dvg. ❆❣♦r❛✱ ♣♦r ✭✷✳✺✮✱ t❡♠♦s H(t)Pn(t) = 1nP′

n(t)✳ ❉❛í

Z

M

(Pn(t) +H(t)Pn(t)hft, νi)dvg =

Z

M

(Pn(t) + 1

nP ′

n(t)hft, νi)dvg = 0. P❡❧❛ ❞❡✜♥✐çã♦ (2.13)✱ t❡♠♦s✿

Z

M

(Pn(t) +H(t)Pn(t)hft, νi)dvg =

Z

M

(20)

✶✶

=

Z

M

(nPn(t)−tPn′(t)hν, νi+P ′

n(t)hf, νi)dvg. ❱❡❥❛ q✉❡ t❡♠♦s✿

Z

M

(nPn(t)−tPn′(t) +Pn′(t)hf, νi)dvg = 0. ✭✸✳✶✮ ❱❛♠♦s ♣r♦✈❛r q✉❡ ✐st♦ é ✉♠ ♣♦❧✐♥ô♠✐♦ ❡♠ t ❝✉❥♦s ❝♦❡✜❝✐❡♥t❡s sã♦ ❛s ✐♥t❡❣r❛✐s ❞❡ ▼✐♥❦♦s✇❦✐✱ ❧♦❣♦ sã♦ ♥✉❧♦s ♦ q✉❡ ❞❡t❡r♠✐♥❛ ❛ ❞❡♠♦♥str❛çã♦✳

❚❡♠♦s✱

Pn(t) = n X r=0 n r

Hrtr

nPn(t) =

n X r=0 n n r Hrtr

Pn′(t) =

n X r=1 n r

rHrtr−1

tPn′(t) =

n X r=0 n r

rHrtr =

n X r=1 n r

n1

r1

rHrtr =n n X r=1

n1

r−1

Hrtr.

❉❛í✱

nPn(t)tPn′(t) =

n X r=0 n n r

Hrtr n

X

r=1

n

n1

r1

Hrtr

=n+

n X r=1 n " n r −

n−1

r1

#

Hrtr.

▲♦❣♦✱

Z

M

(nPn(t)−tPn′(t) +Pn′(t)hf, νi)dvg =

Z M n+ n X r=1 n " n r −

n−1

r1

# Hrtr ! dvg+ Z M n X r=1 n r

rHrtr−1hf, νi ! dvg. ❖❜s❡r✈❡ q✉❡✱ r n r =n

n−1

r1

. ❉❛í✱ = Z M n X r=1 n " n r −

n−1

r1

#

Hrtr+n

! dvg+ Z M n X r=1 n

n−1

r1

Hrtr−1

!

(21)

✶✷

=n

Z

M

(1 +H1hf, νi)dvg +

Z M n X r=1 n " n r −

n1

r1

#

Hrtr

!

dvg+

n−1

X r=1 Z M n

n−1

r

Hr+1hf, νidvg

! tr = n X r=1 Z M n " n r −

n−1

r1

#

Hrdvg

!

tr+

n X r=1 Z M n

n1

r

Hr+1hf, νidvg

!

tr.

P♦rt❛♥t♦✱

0 =

Z

M

(nPn(t)tPn′(t) +Pn′(t)hf, νi)dvg

= n X r=1 Z M n " n r −

n1

r−1

#

Hrdvg

!

tr+

n X r=1 Z M n

n1

r

Hr+1hf, νidvg

!

tr.

❖ q✉❡ ♠♦str❛ q✉❡ t❡♠♦s ✉♠ ♣♦❧✐♥ô♠✐♦ ❡♠ t ❝✉❥♦s ♦s ❝♦❡✜❝✐❡♥t❡s sã♦ ♥✉❧♦s ❡ ❞❛❞♦s ♣♦r✿ 0 = n X r=1 Z M n " n r −

n1

r−1

#

Hrdvg

! + n X r=1 Z M n

n1

r

Hr+1hf, νidvg

! =n Z M " n r −

n−1

r1

#

Hr+

n−1

r

Hr+1hf, νi

! dvg = Z M "

n−1

r

Hr+

n−1

r

Hr+1hf, νi

#

dvg

=

Z

M

(Hr+Hr+1hf, νi)dvg;r = 1, ..., n−1.

❋❛t♦s ✐♠♣♦rt❛♥t❡s✱

nH1+

n

X

r=2

n

n1

r−1

Hrtr−1 =

n−1

X

r=1

n

n1

r

Hr+1+nH1

n r =

n1

r

+

n1

r−1

(22)

✶✸

❚❡♦r❡♠❛ ✸✳✷ ❙❡❥❛f :M −→Rn+1 ✉♠❛ ✐♠❡rsã♦ ❞❡ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛

❡♠ Rn+1 ❝♦♠ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❝♦♥st❛♥t❡ ❡ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ❝♦♥st❛♥t❡✳ ❊♥tã♦ M é ✉♠❛ ❡s❢❡r❛ ❣❡♦❞és✐❝❛✳

❉❡♠♦♥str❛çã♦✿ ❙❛❜❡♠♦s q✉❡ ♣❡❧♦ ❧❡♠❛ (3.1)✱ ♣❛r❛ 0≤r ≤n

Z

M

(Hr1−Hrh(f, ν)i)dvg = 0.

❉❛í✱ t❡♠♦s✿

Z

M

(H0−H1h(f, ν)i)dvg = 0

Z

M

(H1 −H2h(f, ν)i)dvg = 0.

❈♦♠♦ H2 = n(n11)Scal ❡ H1 =H✱ t❡♠♦s q✉❡ H1 ❡H2 sã♦ ❝♦♥st❛♥t❡s✳ ❆ss✐♠✱

Z

M

(H0 −H1h(f, ν)i)dvg =V ol(M)−H1

Z

Mh

(f, ν)idvg =H2−H1H2

Z

Mh

(f, ν)idvg ✭✸✳✷✮ ❡

Z

M

(H1 −H2h(f, ν)i)dvg =H1V ol(M)−H2

Z

M

h(f, ν)idvg =H1H1−H1H2

Z

M

h(f, ν)idvg. ✭✸✳✸✮

❙✉❜tr❛✐♥❞♦ (1) ❞❡ (2)✱ t❡♠♦s❀

H2−H12 = 0

H12 =H2

n

X

i=1

ki2 = 2

n−1

X

i>j kikj.

❖❜s❡r✈❡ q✉❡✱

X

i>j

(ki−kj)2 =X

i>j

(k2i +kj2−2kikj) (n−1)

n

X

i=1

Ki2 −2X

i>j

kikj = 0. ❊♥tã♦✱

kikj = 0ki =kji, j;i > j.

▲♦❣♦✱ M é ✉♠❜í❧✐❝❛ ❡ ♣♦r ❤✐♣ót❡s❡ t❡♠♦s q✉❡ M é ❝♦♠♣❛❝t❛✳ P♦rt❛♥t♦✱ M é ✉♠❛ ❡s❢❡r❛ ❣❡♦❞és✐❝❛✳

▲❡♠❛ ✸✳✸ ❙❡ r 1, ..., n ❡ Hr é ✉♠❛ ❢✉♥çã♦ ♣♦s✐t✐✈❛✱ ❡♥tã♦ H1r

r ≤H

1

r−1

r−1 ≤...≤H

1 2

(23)

✶✹

▲❡♠❛ ✸✳✹ ❙❡ f :M −→Rn+1 é ✉♠❛ ✐♠❡rsã♦ t❛❧ q✉❡ RMf dvg = 0✱❡♥tã♦✿ n 1

vol(M) ≥

Z

M|

f|2dvg. ✭✸✳✹✮

▲❡♠❛ ✸✳✺ ❙❡ f :M −→Rn+1 ❡ r é q✉❛❧q✉❡r ✐♥t❡✐r♦✱ 0≤r ≤n✱ ❡♥tã♦✿

n. 1 V ol(M)

Z

M H2

rdvg ≥λ1(M)

Z

M

Hr−1dvg

2

. ✭✸✳✺✮

◆ós ♦❜t❡r❡♠♦s ❛ ✐❣✉❛❧❞❛❞❡ ♣❛r❛ ❛❧❣✉♠ r✱ 0rn✱ s❡ f ✐♠❡r❣❡ M ❝♦♠♦ ✉♠❛ ❤✐♣❡r❡s❢❡r❛ ❡♠ Rn+1✳

❉❡♠♦♥str❛çã♦✿ ❉❡♥♦t❛r❡♠♦s P ❡ P r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r✿ P := hf, νi ❡ H1 := −P✱ ♦♥❞❡ f é ✐♠❡rsã♦ ✐s♦♠étr✐❝❛ ❡ ν ♦ ❝❛♠♣♦ ✉♥✐tár✐♦ ♥♦r♠❛❧ ❛ M✳

■♥✐❝✐❛❧♠❡♥t❡ ♦❜s❡r✈❡ q✉❡ ✭✸✳✺✮ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❛ ♦r✐❣❡♠ ❞❛❞♦ q✉❡✱ ❛ ú♥✐❝❛ ❡①♣r❡ssã♦ q✉❡ s❡ r❡❧❛❝✐♦♥❛ ❝♦♠ ❛ ❡s❝♦❧❤❛ ❞❛ ♦r✐❣❡♠ é R

MP dvg✱ ❛ q✉❛❧ ❛♣❛r❡❝❡ ❡♠ ✭✸✳✺✮ ♣❛r❛ r = 0 ✭r❡❝♦r❞❡ q✉❡ H1 = −P✮✳ ❉❡ ❢❛t♦✱

R

MP dvg ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❛ ♦r✐❣❡♠✱ ♣♦✐s s❡ tr❛♥s❧❛❞❛r♠♦s f ♣♦r ✉♠ ✈❡t♦r ❝♦♥st❛♥t❡ ❈✱ ❡♥tã♦ ♦❜t❡r❡♠♦s ✉♠❛ ♥♦✈❛ ❢✉♥çã♦✿

P ′

=hf +C, νi=hf, νi+hC, νi=P +hC, νi.

❈♦♠♦ ♣❛r❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❝♦♠♣❛❝t❛s ❡♠ Rn+1✱ RMνdvg = 0✱ t❡♠♦s✿

Z M P ′ = Z M P dvg.

P❡❧❛ ♦❜s❡r✈❛çã♦ ❢❡✐t❛ ❛❝✐♠❛✱ s❡❣✉❡ q✉❡ f ✐♥❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❛ ♦r✐❣❡♠✳ ❉❛✐✱ ♣♦❞❡♠♦s ❛ss✉♠✐r s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ q✉❡ ♦ ❝❡♥tr♦ ❞❛ ❣r❛✈✐❞❛❞❡ ❞❡ f ❡stá ❧♦❝❛❧✐③❛❞♦ ♥❛ ♦r✐❣❡♠✳ ❈♦♠ ✐ss♦✱R

Mf dvg = 0✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ❧❡♠❛(3.4)✱ ♦✉ s❡❥❛✱ ✭✸✳✹✮ ♦❝♦rr❡ ♣❛r❛ 1rn✳ ❉❛í✱

n 1

V ol(M) ≥λ1(M)

Z

M|

f|2dvg.

▼✉❧t✐♣❧✐❝❛♥❞♦ ❝❛❞❛ ❧❛❞♦ ❞❡ ✭✸✳✹✮ ♣♦r (R

MH

2

rdvg)✱ t❡♠♦s✿ n 1

V ol(M)

Z

M

Hr2dvg λ1

Z

M| f|2dvg

! Z

M

Hr2dvg

!

≥λ1

Z

M|

f||Hr||ν|dvg

!2

≥λ1

Z

Mh

f, νidvg

!

✭✸✳✻✮ ❖❜s❡r✈❡ q✉❡ ❛ ú❧t✐♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❡♠ ✭✸✳✻✮✱❢♦✐ ♦❜✐t✐❞❛ ❞❛ ✐♥❡q✉❡çã♦ ❞❡ ❈♦✉❝❤②✲ ❙❝❤✇❛rt③✳ ❆❣♦r❛✱ ♣❡❧❛ ❛ ❢ór♠✉❧❛ ❞❡ ❍s✐✉♥❣✲▼✐♥❦♦✇s❦✐✱ t❡♠♦s✿

λ1(M)

Z

Mh

f, Hrνidvg

!2

=λ1(M)

Z

M

Hr−1dvg

!2

.

(24)

✶✺

n 1 V ol(M)

Z

M H2

rdvg ≥λ1(M)

Z

M

Hr−1dvg

!2

.

P♦t❛♥t♦✱ ♦❜t❡♠♦s ✭✸✳✺✮✳

❆❣♦r❛✱ ♦❜s❡r✈❡ q✉❡ ♣❛r❛ r = 0 t❡♠✲s❡

|f|H0 =|f|.1 =|f|.|ν| ≤ |hf, νi|=|P|.

❉❛í✱

n 1 V ol(M)

Z

M

H0dvg ≤λ1

Z

M| f|2dvg

! Z

M

12dvg

!

≤λ1

Z

M| P|dvg

!2

=λ1(M)

Z

1

H1dvg

!2

.

❆❧é♠ ❞✐ss♦✱ ♦❜s❡r✈❡ q✉❡ q✉❛♥❞♦ ♦❝♦rr❡ ❛ ✐❣✉❛❧❞❛❞❡ ❡♠ ✭✸✳✺✮✱ t❡♠♦s✿

n 1 V ol(M)

Z

M

Hr2dvg =λ1(M)

Z

M | f|2dvg

! Z

M |

Hr|2dvg

!

=λ1(M)

Z

M |

f||H|r|ν|dvg

!2

=λ1(M) hf, Hrνi

!2

.

❉❛í✱ ♣♦r ❈♦✉❝❤②✲❙❝❤✇❛r③ t❡♠♦s q✉❡Hr =cf✱ ♦♥❞❡ ❝ é ✉♠❛ ❝♦♥st❛♥t❡ q✉❛❧q✉❡r✳ ❈♦♠♦✱ ♣♦r ❤✐♣ót❡s❡Hr6= 0✱ t❡♠♦s q✉❡c6= 0✳ ❆ss✐♠✱ ♣♦rHr ❡stá r❡❧❛❝✐♦♥❛❞♦ ❝♦♠ ♦ ✈❡t♦r ♥♦r♠❛❧✱ f é s❡♠♣r❡ ♥♦r♠❛❧ à M✳ ▲♦❣♦✱

d|f|2 = 2hf, dfi= 0,

❞❡ ♦♥❞❡ ❝♦♥❝❧✉✐♠♦s q✉❡ |f| é ❝♦♥st❛♥t❡✳ P♦rt❛♥t♦f ❛♣❧✐❝❛ M ❡♠ ✉♠❛ ❤✐♣❡r❡s❢❡r❛ ❞❡Rn+1✳

❈❛❧❝✉❧❛r❡♠♦s ❛ ❝♦♥st❛♥t❡ kp,r q✉❡ s❡rá ✉t✐❧✐③❛❞❛ ❞✉r❛♥t❡ ❛❧❣✉♥❤❛s ❞❡♠♦♥str❛çõ❡s ❞♦s t❡♦r❡♠❛s ❞❡ r✐❣✐❞❡③ q✉❡ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ♥♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✳ P❛r❛ ❝❛❧❝✉❧❛r♠♦skp,r✱ ✐s♦❧❛♠♦s λ1(M) ♥❛ ❡q✉❛çã♦ ❞♦ ❧❡♠❛ (3.4)✱ t❡♠♦s✿

λ1(M)≤

1 (R

M Hr−1dvg)2 . n

V ol(M).

Z

M

Hr2dvg.

❆❣♦r❛✱ ♦❜s❡r✈❡ q✉❡✿

Z

M

1.Hr2dvg ≤(

Z

M

(Hr2)p)1/p(

Z

M

(1)q)1/q = (

Z

M

Hr2p)2/2p(V ol(M))1/q. ▼❛s✱ 1

q = 1−

1

p✳ ❉❛í✱

Z

M

Hr2dvg ≤ kHrk22p.V ol(M)

1−1

p.

(25)

✶✻

λ1(M)≤

nkHrk22p.V ol(M)−

1/p

(R

MHr−1dvg)2

=kp,r. ❆❣♦r❛✱ ❛♣r❡s❡♥t❛r❡♠♦s ♠❛✐s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s✳ ❉❡✜♥✐çã♦ ✸✳✻ ❉✐③❡♠♦s q✉❡ M é θ✲q✉❛s✐✲✐s♦♠étr✐❝❛ ♣❛r❛ Snq1

k

s❡ ❡①✐st❡ ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧ F ❞❡ M ❡♠ Snq1

k

t❛❧ q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r x M ❡ ♣❛r❛ q✉❛❧q✉❡r ✈❡t♦r ✉♥✐tár✐♦ u∈TxM✱ ♥ós t❡♠♦s✿

||dxF(u)|21| ≤θ, ♦♥❞❡ θ ]0,1[.

❚❡♦r❡♠❛ ✸✳✼ ✭❬✶✸❪✱ ❚❡♦r❡♠❛ ✷✮ ❙❡❥❛ (Mn, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ r✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛✱ ♦r✐❡♥t❛❞❛✱ s❡♠ ❜♦r❞♦✱ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ✐♠❡rs❛ ❡♠ Rn+1✳

❆ss✉♠❛ q✉❡ V ol(M) = 1 ❡ s❡❥❛ r ∈ {1, ..., n} t❛❧ q✉❡ Hr > 0✳ ❊♥tã♦ ♣❛r❛

q✉❛❧q✉❡r p2 ❡ q✉❛❧q✉❡r θ ]0,1[✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ Kθ ❞❡♣❡♥❞❡♥❞♦ s♦♠❡♥t❡ ❞❡ n,kHk,kHrk2p ❡ θ t❛❧ q✉❡ s❡ ❛ ❝♦♥❞✐çã♦ ♣✐♥❝❤✐♥❣✱

(PKθ) 0λ1(M)

Z

M

Hr1dvg

2

− n

V ol(M)1/pkHrk

2

2p >−Kθ

é s❛t✐s❢❡✐t❛✱ ❡♥tã♦ M é ❞✐❢❡♦♠♦r❢❛ ❡ θ✲q✉❛s✐✲✐s♦♠étr✐❝❛ à Snqn λ1

❚❡♦r❡♠❛ ✸✳✽ ✭❬✶✸❪✱ ❈♦r♦❧ár✐♦ ✷✮ ❙❡❥❛ (Mn, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ r✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛✱ ♦r✐❡♥t❛❞❛✱ s❡♠ ❜♦r❞♦✱ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ✐♠❡rs❛ ❡♠ Rn+1, n

3✳❙❡❥❛ θ ]0,1[✳ ❙❡ (Mn, g) é q✉❛s❡✲❡✐♥st❡✐♥✱✐st♦ é✱ kRic (n 1)kgk

∞ ≤ ǫ

♣❛r❛ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛ k✱ ❝♦♠ ǫ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ❞❡♣❡♥❞❡♥❞♦ ❞❡ n, k,kHk∞ ❡ θ✱ ❡♥tã♦ M é ❞✐❢❡♦♠♦r❢❛ ❡ θ✲q✉❛s✐✲✐s♦♠étr✐❝❛ ♣❛r❛ Sn

q

1

k

✳ ❚❡♦r❡♠❛ ✸✳✾ ✭❬✷❪✱ ❚❡♦r❡♠❛ ✶✳✻✮ P❛r❛ q✉❛❧q✉❡r q > n2✱ ❡①✐st❡ C(q, n) t❛❧ q✉❡

s❡ (Mn, g) é ✈❛r✐❡❞❛❞❡ ❝♦♠♣❧❡t❛ ❝♦♠ R

M(Ric −(n− 1))q <

V ol(M)

C(q,n)✱ ❡♥tã♦ M é

❝♦♠♣❛❝t❛✱ t❡♠ ❣r✉♣♦ ❢✉♥❞❛♠❡♥t❛❧ ✜♥✐t♦ ❡ s❛t✐s❢❛③✱

λ1(M)≥n

"

1C(q, n)

ρq V ol(M)

1/q#

♦♥❞❡ ρq =R

M(Ric−(n−1)) q

❉❡✜♥✐çã♦ ✸✳✶✵ ❖ t❡♥s♦r ❞❡ ✉♠❜✐❧✐❝✐❞❛❞❡ é ❞❡✜♥✐❞♦ ♣♦r✿ τ =B −HId,

(26)

✶✼

❚❡♦r❡♠❛ ✸✳✶✶ ✭❬✸❪✱ ❚❡♦r❡♠❛ ✶✳✷✮ ❙❡❥❛ (Mn, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛✱ ♦r✐❡♥t❛❞❛✱ s❡♠ ❜♦r❞♦✱ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ✐♠❡rs❛ ♣♦r f ❡♠ Rn+1✳

❆ss✉♠❛ q✉❡ V(M) = 1 ❡ s❡❥❛ x0 ♦ ❝❡♥tr♦ ❞❡ ♠❛ss❛ ❞❡ M✳ ❊♥tã♦ ♣❛r❛ q✉❛❧q✉❡r

p ≥ 2 ❡ ♣❛r❛ q✉❛❧q✉❡r ǫ > 0 ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ Cǫ ❞❡♣❡♥❞❡♥❞♦ s♦♠❡♥t❡ ❞❡ n, ǫ >0 ❡ ❞❛ L✲♥♦r♠❛ ❞❡ H t❛❧ q✉❡ s❡

(PCǫ) nkHk

2

2p−Cǫ < λ1(M),

❡♥tã♦✱

✶✳ f(M)B

x0,

q n

λ1(M) +ǫ

/B

x0,

q n

λ1(M)−ǫ

✷✳ ∀xS

x0,

q

n λ1(M)

, B(x, ǫ)f(M)6=.

▲❡♠❛ ✸✳✶✷ ✭❬✸❪✱ ▲❡♠❛ ✶✳✶✮ ❙❡❥❛ (Mn, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ (n 2) ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛✱ ♦r✐❡♥t❛❞❛✱ s❡♠ ❜♦r❞♦ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ✐♠❡rs❛ ♣♦r f ❡♠ Rn+1

❆ss✉♠❛ q✉❡ ♦ V(M) = 1✳ ❊♥tã♦ ❡①✐st❡ ❝♦♥st❛♥t❡ cn ❡ dn ❞❡♣❡♥❞❡♥❞♦ s♦♠❡♥t❡ ❞❡ n t❛❧ q✉❡ s❡ ♣❛r❛ q✉❛❧q✉❡rp≥2✱s❡ (PC) é ✈❡r❞❛❞❡ ❝♦♠ C < cn✱ ❡♥tã♦✿

n λ1(M) ≤

dn.

▲❡♠❛ ✸✳✶✸ ✭❬✸❪✱ ▲❡♠❛ ✹✳✶✮ P❛r❛ p 2 ❡ ♣❛r❛ q✉❛❧q✉❡r η>0✱ ❡①✐st❡

Kη(n,kBk) t❛❧ q✉❡ s❡ (Pkη)é ✈❡r❞❛❞❡✐r❛✱ ❡♥tã♦ kψk∞ ≤η✳ ❆❧é♠ ❞✐ss♦✱ Kη →0

q✉❛♥❞♦ kBk∞→ ∞ ♦✉ η→0✳

❚❡♦r❡♠❛ ✸✳✶✹ ❙❡❥❛ (Mn, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ♥✲❞✐♠❡♥s✐♦♥❛❧ (n2) ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛✱ ♦r✐❡♥t❛❞❛✱ s❡♠ ❜♦r❞♦✱ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ✐♠❡rs❛ ♣♦r f ❡♠ Rn+1

❆ss✉♠❛ q✉❡ V(M) = 1 ✳ ❊♥tã♦ ♣❛r❛ q✉❛❧q✉❡r p 2 ✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ C ❞❡♣❡♥❞❡♥❞♦ s♦♠❡♥t❡ ❞❡ n ❡ ❞❛ L✲♥♦r♠❛ ❞❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ B t❛❧ q✉❡ s❡

(PC) nkHk2

2p−C < λ1(M),

❡♥tã♦ M é ❞✐❢❡♦♠♦r❢❛ ♣❛r❛ Sn

q

n λ1(M)

❉❡♠♦♥str❛çã♦✿ Pr✐♠❡✐r♦ t♦♠❡

ǫ < 1

2

r

n

kBk

r

n λ1(M)

. ✭✸✳✼✮

❈♦♠ ❡st❛ ❡s❝♦❧❤❛ ❞❡ǫ♥ós t❡♠♦s q✉❡ s❡ ❛ ❝♦♥❞✐çã♦ ♣✐♥❝❤✐♥❣ é ✈❡r❞❛❞❡✐r❛✱ ❡♥tã♦

|Xx| ♥✉♥❝❛ s❡ ❛♥✉❧❛✱ ♣♦✐s s❡ PCǫ é ✈á❧✐❞❛ ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛(3.11) q✉❡

❡stá ♣r♦✈❛❞♦ ♥❛ r❡❢❡rê♥❝✐❛ ❬✸❪✱ ❢♦✐ ♣r♦✈❛❞♦ q✉❡ ✈❛❧❡✿

||Xx| − r

n λ1(M)| ≤

ǫ⇒ r

n λ1(M) −

ǫ ≤ |X| ≤ r

n λ1(M)

(27)

✶✽

❉❡ ✭✸✳✼✮ t❡♠♦s✱

ǫ <

r n

λ1(M) ⇒ −

ǫ >

r n

λ1(M)

.

❉❛í ❞❛ ❡q✉❛çã♦ (3.8)t❡♠♦s✱

r n

λ1(M) −

r n

λ1(M)

<|Xx|<2

r n

λ1(M)

0<|Xx|<2

r n

λ1(M)

.

▲♦❣♦✱ |Xx| ♥✉♥❝❛ s❡ ❛♥✉❧❛✳

❈♦♠ ✐ss♦ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❛ ❛♣❧✐❝❛çã♦ ❞✐❢❡r❡♥❝✐á✈❡❧ F ❞❛❞❛ ♣♦r✿

F :M S

0,

r n

λ1(M)

x

r n

λ1(M)

Xx

|Xx|.

❆ q✉❛❧ ✐r❡♠♦s ♣r♦✈❛r q✉❡ é ✉♠❛ q✉❛s✐ ✐s♦♠❡tr✐❛✳ ❉❡ ❢❛t♦✱ ♥ós ✐r❡♠♦s ♠♦str❛r q✉❡ ♣❛r❛ q✉❛❧q✉❡r 0< θ < 1✱ ♥ós ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ✉♠❛ ❝♦♥st❛♥t❡ ǫ(n,kBk∞, θ)

t❛❧ q✉❡ ♣❛r❛ q✉❛❧q✉❡r x M ❡ q✉❛❧q✉❡r ✈❡t♦r ✉♥✐tár✐♦ u TxM✱ ❛ ❝♦♥❞✐çã♦ ♣✐♥❝❤✐♥❣ PCǫ(n,kBk∞,θ) ✐♠♣❧✐❝❛✱

||dFx(u)|2−1| ≤θ

Pr✐♠❡✐r❛♠❡♥t❡✱ ✐r❡♠♦s ❝❛❧❝✉❧❛r |dFx(u)|2✳ ❖❜s❡r✈❡ q✉❡✱

dFx(u) =

q

n λ1(M)∇

0

u

X

|X|

x✱ ♦♥❞❡∇

0 é ❝♦♥❡①ã♦ ❘✐❡♠❛♥♥✐❛♥❛ ❞♦

Rn+1✳ ❆ss✐♠

dFx(u) =

r n

λ1(M)∇ 0

u

X

|X|

x

dFx(u) =

r n

λ1(M)

u

1

|X|

X+ 1

|X|∇

0 uX = r n λ1(M)

−1

2

u(|X|2)

|X|3 X+

1

|X|

=

r n

λ1(M)

1

|X|3h∇ 0

uX, XiX+

1

|X|.u

=

r n

λ1(M)

1

|X|

− 1

|X|2hu, XiX+u

❉❛í✱

|dFx(u)|2 = n

λ1(M)

1

|X|2 1−

hu, Xi2 |X|2

!

(28)

✶✾

❈♦♠ ✐ss♦ t❡♠♦s✿

||dx(u)|2−1|=| n λ1(M)

1

|X|2 1−

hu, Xi2

|X|2 −1

!

| ≤ | n

λ1(M)

1

|X|2−1|+

n λ1(M)

1

|X|4hu, Xi.

✭✸✳✾✮ ❆❣♦r❛✱ ♥♦t❡ q✉❡

| n

λ1(M)

1

|X|2 −1|=

1

|X|2|

n λ1(M)− |

X|2|

1

|X|2|

r n

λ1(M) − |

X|

||

r n

λ1(M)

+|X|

|

♣♦r s❛❜❡♠♦s q✉❡✱

r

n λ1(M)−

ǫ≤ |X| ≤ r

n λ1(M)

+ǫ. ❉❛í✱

r n

λ1(M)− ≤

ǫ.

❊♥tã♦✱

| n

λ1(M)

1

|X|2 −1| ≤

ǫ|

q

n

λ1(M) +|X|

|

|X|2

❛✐♥❞❛ ❞❡ ✐r❡♠♦s ♦❜t❡r

r

n λ1(M)

+|X|

≤2

r

n λ1(M)

+ǫ ❡

r n

λ1(M)−

ǫ

2

≤ |X|2.

▲♦❣♦✱

|λ n

1(M)

1

|X|2 −1| ≤

ǫ|

q

n

λ1(M) +|X|

|

|X|2

≤ǫ

2q n λ1(M)+ǫ

q n

(29)

✷✵

❉♦ ❧❡♠❛ (3.12)✱ t❡♠♦s dnn ≤ λ1(M) ≤ kBk2✳ ❈♦♠♦ ♥ós ❛ss✉♠✐♠♦s ǫ < 1

2

q n

kBk∞✱ ♦ ❧❛❞♦ ❞✐r❡✐t♦ é ❧✐♠✐t❛❞♦ ❛❝✐♠❛ ♣❡❧❛ ❝♦♥st❛♥t❡ ❞❡♣❡♥❞❡♥❞♦ s♦♠❡♥t❡ ❞❡ n ❡ ❞❡ kBk∞✳ ▲♦❣♦✱ t❡r❡♠♦s✿

| n

λ1(M)

1

|X|2 −1| ≤ǫγ(n,kBk∞). ✭✸✳✶✵✮

P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❡s❞❡ q✉❡Cǫ(n,kBk)0q✉❛♥❞♦ǫ 0✱ ❡①✐st❡ǫ(n,kBk, η)

t❛❧ q✉❡ Cǫ(n,kBk∞,η) ≤ Kη(n,kBk∞) ✭♦♥❞❡ Kη é ❛ ❝♦♥st❛♥t❡ ❞♦ ❧❡♠❛ (3.13)✮ ❡ ❡♥tã♦ ♣❡❧♦ ❧❡♠❛ (3.13)✱ kψk2

∞≤η2✳ ❆ss✐♠✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡δ❞❡♣❡♥❞❡♥❞♦ s♦♠❡♥t❡

❞❡n ❡kBk t❛❧ q✉❡✱ n λ1(M)

1

|X|4hu, Xi 2

≤ n

λ1(M)

1

|X|4kψ 2

∞k ≤η2δ(n,kBk∞). ✭✸✳✶✶✮

❊♥tã♦✱ ❞❡ ✭✸✳✾✮✱ ✭✸✳✶✵✮ ❡ ✭✸✳✶✶✮ ♥ós ❞❡❞✉③✐♠♦s q✉❡ ❛ ❝♦♥❞✐çã♦ PCǫ(n,kBk∞,η) ✐♠♣❧✐❝❛

||dFx(u)|21| ≤ǫγ(n,kBk) +η2δ(n,kBk). ❆❣♦r❛✱ ♥ós ❡s❝♦❧❤❡♠♦s η = 2θδ1/2

✳ ❊♥tã♦✱ ♥ã♦ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ ǫ(n,kBk, η) é s✉✜❝✐❡♥t❡ ♣❡q✉❡♥♦ ♥❛ ♦r❞❡♠ ♣❛r❛ t❡r ǫ(n,kBk, η)γ(n,kBk)

θ

2✳ ◆❡st❡ ❝❛s♦ ♥ós t❡♠♦s✿

||dFx(u)|21| ≤θ.

❆❣♦r❛✱ ♥ós ✜①❛♠♦s θ✱ 0< θ <1✳ ❙❡❣✉❡ q✉❡ F é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❧♦❝❛❧ ❞❡ M ♣❛r❛ Sn

q n

λ1(M)

✳ ❉❡s❞❡ q✉❡Sn

q n

λ1(M)

é s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①❛ ♣❛r❛ n≥2✱

F é ✉♠ ❞✐❢❡♦♠♦r✜s♠♦ ❣❧♦❜❛❧✳

❚❡♦r❡♠❛ ✸✳✶✺ ✭❬✶✶❪✱ ❚❡♦r❡♠❛ ✷✮ ❙❡❥❛ Mn ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❝♦♠♣❛❝t❛ ♠❡r❣✉❧❤❛❞❛ ♥♦ ❡s♣❛ç♦ ❡✉❝❧✐❞✐❛♥♦Rn+1✳ ❙❡ Hr é ❝♦♥st❛♥t❡ ♣❛r❛ ❛❧❣✉♠ r= 1, ..., n✱

(30)

❈❛♣ít✉❧♦ ✹

❚❊❖❘❊▼❆❙ ❉❊ ❘■●■❉❊❩

◆❡st❡ ❝❛♣ít✉❧♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ t❡♦r❡♠❛ ❞❡ r✐❣✐❞❡③ ♣❛r❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡s ❡♠

Rn+1 ♥❛ ❝❧❛ss❡ ❞❛s ❤✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲❊✐♥st❡✐♥ q✉❡ ❣❡♥❡r❛❧✐③❛ ♦ t❡♦r❡♠❛ ✭✸✳✽✮

❡♥✉♥❝✐❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ✸✳ ❈♦♠♦ ❛♣❧✐❝❛çõ❡s ❞❡st❡s t❡♦r❡♠❛s✱ ♣r♦✈❛r❡♠♦s r❡s✉❧t❛❞♦s s✐♠✐❧❛r❡s ♣❛r❛ ❛ ❝❧❛ss❡ ❞❛s ❤✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲✉♠❜í❧✐❝❛s✱ q✉❡ ♣♦s✐❜✐❧✐t❛ ❣❡r❛r ❢❡rr❛♠❡♥t❛s✱ ❝♦♠♦ ♦ t❡♦r❡♠❛ ✭✹✳✺✮✱ q✉❡ s❡rã♦ ✉t✐❧✐③❛❞❛s ♥❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧✳ ❊♥❝❡r❛r❡♠♦s ❡st❡ ❝❛♣ít✉❧♦ ❝♦♠ ❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❡ ✉♠ ❝♦r♦❧ár✐♦✳

✹✳✶ ❍✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲❊✐♥st❡✐♥

❖ r❡s✉❧t❛❞♦ ❛❜❛✐①♦✱ ❢♦✐ ❡♥✉♥❝✐❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ♣♦ré♠ ❝♦♠ ❛ ♥♦r♠❛

k.k✱ ❡ ♣r♦✈❛❞♦ ♥❛ r❡❢❡rê♥❝✐❛ ❬✶✸❪✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ❝♦♥s✐❞❡r❛♠♦s ❤❡✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲❊✐♥st❡✐♥ ❞❡ Rn+1 ❡♠ ✉♠ s❡♥t✐❞♦ ❢r❛❝♦✱ ✐st♦ é✱ s❛t✐❢❛③❡♥❞♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡

kRic−(n−1)kp,rgkq ≤ǫ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛k ❡ ♣❛r❛ǫs✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✳ Pr❡❝✐s❛♠❡♥t❡✱ ♣r♦✈❛♠♦s ♦ s❡❣✉✐♥t❡✿

❚❡♦r❡♠❛ ✹✳✶ ❙❡❥❛ (Mn, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ r✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛✱ ♦r✐❡♥t❛❞❛✱ s❡♠ ❜♦r❞♦✱ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ✐♠❡rs❛ ❡♠Rn+1✳❙❡❥❛θ ∈]0,1[✳ ❙❡(Mn, g) é q✉❛s❡✲❡✐♥st❡✐♥✱ ✐st♦ é✱ kRic(n1)kp,rgkq ≤ ǫ ♣❛r❛ ❛❧❣✉♠ ǫ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ❞❡♣❡♥❞❡♥❞♦ ❞❡n, k,kHk ❡θ✱ ❡♥tã♦M é ❞✐❢❡♦♠♦r❢❛ ❡θ✲q✉❛s✐✲✐s♦♠étr✐❝❛ ♣❛r❛ Snq 1

kp,r

.

❉❡♠♦♥str❛çã♦✿

❙❛❜❡♠♦s q✉❡ s❡ kRic−(n−1kg)kq ≤λ(n, q, k) ♣❛r❛ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛✱ k ❝♦♠ q > n

2 ❡ ǫ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ t❡♠♦s q✉❡ ♣❡❧♦ t❡♦r❡♠❛ (3.9) λ1(M)

s❛t✐s❢❛③✱

λ1(M)≥nk(1−Cǫ), ♦♥❞❡ Cǫ é ✉♠❛ ❝♦♥st❛♥t❡ t❛❧ q✉❡ Cǫ →0q✉❛♥❞♦ ǫ→0✳

❆❣♦r❛✱ t♦♠❡ k =kp,r✳ ◆ós ♦❜t❡♠♦s❀

(31)

✷✷

λ1(M)≥

n2kHrk2

2pV ol(M)−1/p

R

MHr−1dvg

2 (1−Cǫ)

λ1(M)

Z

M

Hr1dvg

2

− nkHrk

2 2p V ol(M)1/p >−

nkHrk2 2p V ol(M)1/pCǫ, t♦♠❡ Kǫ = nkHrk22p

V ol(M)1/pCǫ✳ ❉❛í✱

λ1(M)

Z

M

Hr1dvg

2

− nkHrk

2 2p

V ol(M)1/p >−Kǫ.

▲♦❣♦✱ ♣❛r❛θ ∈]0,1[✱ ❡s❝♦❧❤❡♠♦sǫ(n, q, k, θ)s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ t❛❧ q✉❡Kǫ é s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ❡♥tã♦ ❞♦ t❡♦r❡♠❛ (3.7) ♦❜t❡♠♦s q✉❡M é ❞✐❢❡♦♠♦r❢❛ ❡ θ✲q✉❛s✐✲✐s♦♠étr✐❝❛ ♣❛r❛ Snq 1

kp,r

❉♦ t❡♦r❡♠❛ ✭✹✳✶✮✱ ♥ós ❞❡❞✉③✐r❡♠♦s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ♣❛r❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲✉♠❜í❧✐❝❛s ❞♦Rn+1✳

✹✳✷

❍✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲❯♠❜í❧✐❝❛s

Pr✐♠❡✐r♦✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ♦ q✉❛❧ é ✉♠❛ ❛♣❧✐❝❛çã♦ ❞✐r❡t❛ ❞♦ ❚❡♦r❡♠❛ ✭✸✳✽✮✳

❚❡♦r❡♠❛ ✹✳✷ ❙❡❥❛ (Mn, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ r✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛✱ ♦r✐❡♥t❛❞❛✱ s❡♠ ❜♦r❞♦✱ ✐s♦♠❡tr✐❝❛♠❡♥t❡ ✐♠❡rs❛ ❡♠ Rn+1✳ ❙❡❥❛ θ ]0,1[✳ ❙❡ (Mn, g) é q✉❛s❡✲✉♠❜í❧✐❝❛✱ ✐st♦ é✱ kB kgk

∞ ≤ ǫ ♣❛r❛ ✉♠❛ ❝♦♥st❛♥t❡ ♣♦s✐t✐✈❛

k✱ ❝♦♠ǫ s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ ❞❡♣❡♥❞❡♥❞♦ ❞❡ n, k ❡ θ✱ ❡♥tã♦M é ❞✐❢❡♦♠♦r❢❛ ❡ θ✲q✉❛s✐✲✐s♦♠étr✐❝❛ à Sn 1

k

✳ ❉❡♠♦♥str❛çã♦✿

P❛r❛ ♠♦str❛r q✉❡M é ❞✐❢❡♦♠♦r✜❝❛ ❡θ✲q✉❛s✐✲✐s♦♠étr✐❝❛ àSn 1

k

✱ ♠♦str❛r❡♠♦s q✉❡ (Mn, g) ❝♦♠ ❛s ❤✐♣ót❡s❡s ❞♦ t❡♦r❡♠❛(4.2) é q✉❛s❡✲❡✐♥st❡✐♥✱ ❞❛í ♣❡❧♦ t❡♦r❡♠❛ ✭✸✳✽✮ ❞❡t❡r♠✐♥❛✲s❡ ❛ ❞❡♠♦♥tr❛çã♦✳

Pr✐♠❡✐r♦✱ ♥♦t❡ q✉❡❀

Ric(Y, Y) =nHhB(Y), Yi − hB(Y), B(Y)i ✭✹✳✶✮

✈❛❧❡ ♣❛r❛ ✉♠❛ ❜❛s❡ {ei} ♦rt♦♥♦r♠❛❧ ❞❡ TpM✳ ❉❡ ❢❛t♦✱

❙❡❥❛ f :Mn−→Rn+1 ✉♠❛ ✐♠❡rsã♦✱ ❡♥tã♦ ❞❛❞♦pM ν (TpM) |ν|= 1

❈♦♠♦ Sν : TpM −→ TpM é s✐♠étr✐❝❛✱ ❡①✐st❡ ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ ✈❡t♦r❡s ♣ró♣r✐♦s{e1, ..., en}❞❡TpM ❝♦♠ ✈❛❧♦r❡s ♣ró♣r✐♦s r❡❛✐sλ1, ..., λn✱ ✐✳❡✱Sν(ei) = λiei❀

1in✳ P❡❧❛ ❡q✉❛çã♦ ❞❡ ●❛✉ss✱ t❡♠♦s q✉❡✿

(32)

✷✸

X

k

hR(ei, ek)ei, eki=X

k

hB(ek, ek), B(ei, ei)i −X

k

hB(ei, ek), B(ek, ei)i

Ric(ei, ei) =X

k

hB(ek, ek), B(ei, ei)i −X

k

hB(ei, ek), B(ek, ei)i. ❆♥❛❧✐s❛r❡♠♦s s❡♣❛r❛❞❛♠❡♥t❡ ♦s ❞♦✐s s♦♠❛tór✐♦s✿

1◦ PkhB(ek, ek), B(ei, ei)i.

❖❜s❡r✈❡ q✉❡ B(x, y) = hB(x), yi ⇒B(x) =xν =Sν(x)✳ ❆❧é♠ ❞✐ss♦✱

hB(ek, ek), B(ei, ei)i=hSB(ei,ei)(ek), eki

=hB(ei, ei, νihSν(ek), eki. ❊♥tã♦✱

X

k

hB(ek, ek), B(ei, ei)i=hB(ei, ei), νiX k

hSν(ek), eki

=hB(ei, ei), νinH =hSν(ei), eiinH =hB(ei), eiinH. 2◦ P

khB(ei, ek), B(ek, ei)i.

hB(ei, ek), B(ek, ei)i=hB(ek, ei), νihSν(ei), eki

=hSν(ek), eiihSν(ei), eki

=hλkek, eiihλiei, eki

=λkλiδki2. ▲♦❣♦✱

X

k

hB(ei, ek), B(ek, ei)i=λiX k

λkδ2ki =λiλi

=λiλihei, eii=hλiei, λieii

=hSν(ei), Sν(ei)i=hB(ei), B(ei)i. ❉❛í✱

Ric(ei, ei) =nHhB(ei), eii − hB(ei), B(ei)i.

Referências

Documentos relacionados

Em entrevista ao jornal Pioneiros a artista relata essa questão.. No registro do vídeo é possível acompanhar, com mais precisão, a agitação dos anaimais em

Em locais com características que priorizam os pedestres podem encorajar as viagens realizadas a pé, mostrando que o ambiente construído pode interferir na opção

Por meio do uso da fase estacionária quiral MCTA (triacetato de meti! celulose) é possível á separação dos enantiômeros da mistura racêmica do Rolipram Com

Nos olhos foi aplicado primer para fixação e coloração melhor, no canto interno foi aplicado um tom de branco na pálpebra móvel foram aplicadas as seguintes cores

Com a crise energética dos anos 70, ocasionada pela aumento do preço do petróleo, surge a necessidade de se repensar a tipologia do edifício de escritório, surgindo, a partir do

A Presidente do Conselho Municipal de Assistência Social do Município de Balsa Nova, no uso de suas atribuições conferidas na Lei Municipal n° 288/1995, alterada pela Lei Municipal

Não podemos deixar de lembrar que “ assim também nós, embora muitos, somos um só corpo em Cristo e, individualmente, membros uns dos outros ” (Romanos 12.5).. Para o

Outras pesquisas poderiam ser efetuadas com intuito de abordar a relação entre os tipos de ataque que originaram os gols, o tempo de jogo e suas respectivas distâncias da