❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❖ ❈❊❆❘➪
❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙
❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆
P❘❖●❘❆▼❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼
▼❆❚❊▼➪❚■❈❆
◆❊■▲❍❆ ▼❆❘❈■❆ P■◆❍❊■❘❖
❘■●■❉❊❩ ❉❆ ❊❙❋❊❘❆ ◆❖ ❊❙P❆➬❖ ❊❯❈▲■❉■❆◆❖
◆❊■▲❍❆ ▼❆❘❈■❆ 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❛✳
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✳
◆❡✐❧❤❛ ▼❛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 ✭❯❋❈✮
❆●❘❆❉❊❈■▼❊◆❚❖
❆ ❉❡✉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❛♠ ❝♦♠✐❣♦✳
❘❊❙❯▼❖
◆❡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✳
❆❇❙❚❘❆❈❚
■♥ 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✳
❙❯▼➪❘■❖
✶ ■◆❚❘❖❉❯➬➹❖ ✶
✷ P❘❊▲■▼■◆❆❘❊❙ ✷
✸ ❋❆❚❖❙ ❇➪❙■❈❖❙ ✶✵
✹ ❚❊❖❘❊▼❆❙ ❉❊ ❘■●■❉❊❩ ✷✶
✹✳✶ ❍✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲❊✐♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✹✳✷ ❍✐♣❡rs✉♣❡r❢í❝✐❡s q✉❛s❡✲❯♠❜í❧✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✹✳✸ Pr♦✈❛ ❞♦ ❚❡♦r❡♠❛ Pr✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
❈❛♣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
✷✳ |Scal−s| ≤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✐♥❝✐♣❛❧✳
❈❛♣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✿
✸
❉❡♠♦♥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✿
✹
∇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♦✱♦♥❞❡✿
✺
❉❡✜♥✐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.
✻
❉❡✜♥✐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
= 2hek−ekhf, 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
✼
❉❡♠♦♥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❡❛❧p≥0✱ ♥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 g∈L1(R) ❡ kf.gk
✽
❙❡❥❛ 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 Sη◦Sζ−Sζ◦Sη ✳
❆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✈❡❧✳
✾
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✳
❈❛♣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
(Hr−1−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
✶✶
=
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
n−1
r−1
rHrtr =n n X r=1
n−1
r−1
Hrtr.
❉❛í✱
nPn(t)−tPn′(t) =
n X r=0 n n r
Hrtr− n
X
r=1
n
n−1
r−1
Hrtr
=n+
n X r=1 n " n r −
n−1
r−1
#
Hrtr.
▲♦❣♦✱
Z
M
(nPn(t)−tPn′(t) +Pn′(t)hf, νi)dvg =
Z M n+ n X r=1 n " n r −
n−1
r−1
# Hrtr ! dvg+ Z M n X r=1 n r
rHrtr−1hf, νi ! dvg. ❖❜s❡r✈❡ q✉❡✱ r n r =n
n−1
r−1
. ❉❛í✱ = Z M n X r=1 n " n r −
n−1
r−1
#
Hrtr+n
! dvg+ Z M n X r=1 n
n−1
r−1
Hrtr−1
!
✶✷
=n
Z
M
(1 +H1hf, νi)dvg +
Z M n X r=1 n " n r −
n−1
r−1
#
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
r−1
#
Hrdvg
!
tr+
n X r=1 Z M n
n−1
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 −
n−1
r−1
#
Hrdvg
!
tr+
n X r=1 Z M n
n−1
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 −
n−1
r−1
#
Hrdvg
! + n X r=1 Z M n
n−1
r
Hr+1hf, νidvg
! =n Z M " n r −
n−1
r−1
#
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
n−1
r−1
Hrtr−1 =
n−1
X
r=1
n
n−1
r
Hr+1+nH1
❡ n r =
n−1
r
+
n−1
r−1
✶✸
❚❡♦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
(Hr−1−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(n1−1)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♦✱
ki−kj = 0⇒ki =kj∀i, 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
✶✹
▲❡♠❛ ✸✳✹ ❙❡ 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✱ 0≤r≤n✱ s❡ f ✐♠❡r❣❡ M ❝♦♠♦ ✉♠❛ ❤✐♣❡r❡s❢❡r❛ ❡♠ Rn+1✳
❉❡♠♦♥str❛çã♦✿ ❉❡♥♦t❛r❡♠♦s P ❡ −P r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r✿ P := hf, νi ❡ H−1 := −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✉❡ H−1 = −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❛ 1≤r≤n✳ ❉❛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
.
✶✺
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
H−1dvg
!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.
✶✻
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)|2−1| ≤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 p≥2 ❡ q✉❛❧q✉❡r θ ∈]0,1[✱ ❡①✐st❡ ✉♠❛ ❝♦♥st❛♥t❡ Kθ ❞❡♣❡♥❞❡♥❞♦ s♦♠❡♥t❡ ❞❡ n,kHk∞,kHrk2p ❡ θ t❛❧ q✉❡ s❡ ❛ ❝♦♥❞✐çã♦ ♣✐♥❝❤✐♥❣✱
(PKθ) 0≥λ1(M)
Z
M
Hr−1dvg
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
"
1−C(q, n)
ρq V ol(M)
1/q#
♦♥❞❡ ρq =R
M(Ric−(n−1)) q✳
❉❡✜♥✐çã♦ ✸✳✶✵ ❖ t❡♥s♦r ❞❡ ✉♠❜✐❧✐❝✐❞❛❞❡ é ❞❡✜♥✐❞♦ ♣♦r✿ τ =B −HId,
✶✼
❚❡♦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
✷✳ ∀x∈S
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✐♦♥❛❧ (n≥2) ❝♦♠♣❛❝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)
✶✽
❉❡ ✭✸✳✼✮ 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
!
✶✾
❈♦♠ ✐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
✷✵
❉♦ ❧❡♠❛ (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)|2−1| ≤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)|2−1| ≤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✱
❈❛♣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−(n−1)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❀
✷✷
λ1(M)≥
n2kHrk2
2pV ol(M)−1/p
R
MHr−1dvg
2 (1−Cǫ)
λ1(M)
Z
M
Hr−1dvg
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
Hr−1dvg
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♦ ❞❛❞♦p∈M ❡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❀
1≤i≤n✳ P❡❧❛ ❡q✉❛çã♦ ❞❡ ●❛✉ss✱ t❡♠♦s q✉❡✿
✷✸
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.