❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
■❞❡♥t✐❞❛❞❡s ●❡♦♠étr✐❝❛s ❡ ❘❡s✉❧t❛❞♦s ❞❡
❘✐❣✐❞❡③ ❡♠ ❱❛r✐❡❞❛❞❡s ❞♦ ❚✐♣♦ ❊stát✐❝❛
♣♦r
❆❧❧❛♥ ●❡♦r❣❡ ❞❡ ❈❛r✈❛❧❤♦ ❋r❡✐t❛s
❇❡❧♦ ❍♦r✐③♦♥t❡
❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
■❞❡♥t✐❞❛❞❡s ●❡♦♠étr✐❝❛s ❡ ❘❡s✉❧t❛❞♦s ❞❡ ❘✐❣✐❞❡③ ❡♠
❱❛r✐❡❞❛❞❡s ❞♦ ❚✐♣♦ ❊stát✐❝❛
♣♦r
❆❧❧❛♥ ●❡♦r❣❡ ❞❡ ❈❛r✈❛❧❤♦ ❋r❡✐t❛s
∗❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡
❉❖❯❚❖❘ ❊▼ ▼❆❚❊▼➪❚■❈❆
❇❡❧♦ ❍♦r✐③♦♥t❡✱ ✵✽ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✶✻
❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛✿
❊③❡q✉✐❡❧ ❘♦❞r✐❣✉❡s ❇❛r❜♦s❛ ✭❖r✐❡♥t❛❞♦r✮
❊♠❡rs♦♥ ❆❧✈❡s ▼❡♥❞♦♥ç❛ ❞❡ ❆❜r❡✉
▼ár❝✐♦ ❍❡♥r✐q✉❡ ❇❛t✐st❛ ❞❛ ❙✐❧✈❛
▼❛r❝♦s ❞❛ ❙✐❧✈❛ ▼♦♥t❡♥❡❣r♦
▼❛r❝♦s P❡trú❝✐♦ ❞❡ ❆❧♠❡✐❞❛ ❈❛✈❛❧❝❛♥t❡
❘♦❞♥❡② ❏♦s✉é ❇✐❡③✉♥❡r
❆❣r❛❞❡❝✐♠❡♥t♦s
❙❡♠ ❞ú✈✐❞❛s✱ ✉♠❛ ❣r❛♥❞❡ r❡❛❧✐③❛çã♦ ❡♠ ♥♦ss❛ ✈✐❞❛ ♣❡r♣❛ss❛ ♣❡❧♦s ♥✉♠❡r♦s♦s ❡s❢♦rç♦s q✉❡ ❝♦♠♣õ❡♠ ❛s ✈✐tór✐❛s ❞✐ár✐❛s✱ ❧❛♠♣❡❥♦s q✉❡ ❝♦♥str♦❡♠ ❛ ♣❛ss♦s ❞❡ ❧✉t❛ ♦ r❡❝♦♥❤❡❝✐✲ ♠❡♥t♦ ❞❛ ❣ê♥❡s❡ ❞❡ ✉♠ s♦♥❤♦✳ ●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛ t♦❞♦s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❞✐r❡t❛ ❡ ✐♥❞✐r❡t❛♠❡♥t❡ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡ ♠❡✉ ❞♦✉t♦r❛❞♦✱ q✉❡ t❡✈❡ ❝♦♠♦ ♣♦♥t♦ ✜♥❛❧ ❡st❛ t❡s❡✳
Pr✐♠♦r❞✐❛❧♠❡♥t❡ ❛ ❉❡✉s✱ ✉♥♦ ❡ tr✐♥♦✱ ❢♦♥t❡ ❡ ✜♠ ❞❡ t♦❞❛s ❛s ❣r❛ç❛s ❞❡♣♦s✐t❛❞❛s ✐♥t❡❣r❛❧♠❡♥t❡ ❞✉r❛♥t❡ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❡st❛ ❢❛s❡ ❡♠ ♠✐♥❤❛ ✈✐❞❛✱ ❝✉❥❛ ♣❛③✱ s❛ú❞❡✱ ❢é ❡ ♣❡rs❡✈❡r❛♥ç❛ ❢♦r❛♠ ❞♦♥s ❝♦♥❝❡❞✐❞♦s ❡ s❡♥t✐❞♦s ❞✐❛r✐❛♠❡♥t❡✳
➚ ▼ã❡ ❞❡ ❉❡✉s ❡ ♥♦ss❛✱ ◆♦ss❛ ❙❡♥❤♦r❛✱ ❱✐r❣❡♠ ❞❛s ●r❛ç❛s ❡ ▼ã❡ ❆♣❛r❡❝✐❞❛✱ tã♦ ♣r❡s❡♥t❡ ❡ ❢❛③❡♥❞♦✲♠❡ s❡♥t✐r✲s❡ ❝✉✐❞❛❞♦✱ ♠❡s♠♦ ✜s✐❝❛♠❡♥t❡✱ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❡ ❡♠ t♦❞❛s ♠✐♥❤❛s ❛çõ❡s✳ ❈♦♠♦ ❡♠ ♠✐♥❤❛s ♦r❛çõ❡s ❞✐ár✐❛s✱ ✧♥ã♦ ❞❡s♣r❡③♦✉ ❛s sú♣❧✐❝❛s q✉❡ ❡♠ ♥♦ss❛ ♥❡❝❡ss✐❞❛❞❡ ✈♦s ❞✐r✐❣✐✱ ♠❛s ❧✐✈r♦✉✲♠❡ s❡♠♣r❡ ❞❡ t♦❞♦s ♦s ♣❡r✐❣♦s✱ ó ❱✐r❣❡♠✱ ❣❧♦r✐♦s❛ ❡ ❇❡♥❞✐t❛✧✳ P❡❧❛s ♠ã♦s ❞❡ ▼❛r✐❛✱ ♦❢❡r❡ç♦✱ s❡♠ r❡s❡r✈❛s✱ ❡st❡ tr❛❜❛❧❤♦ ❡ t♦❞♦s ♦s ❡s❢♦rç♦s ❞✐s♣❡♥❞✐❞♦s ♣❡❧❛ ❣❧ór✐❛ ❞❡ ❉❡✉s✳
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❛ ❢♦♥t❡ q✉❡ ♥ã♦ ✜♥❞❛ ❡♠ ♣r❡s❡♥ç❛ ❞❡ ❛♠♦r ❡ ❛♣♦✐♦ ❡♠ t♦❞♦s ❡ss❡s ❛♥♦s✳ ❙❡✐ q✉❡ ❝♦♥❝❧✉♦ ✉♠ s♦♥❤♦ ❞❡ ♠✉✐t♦s ♣❛ss♦s ❡ ❛ ✈✐tór✐❛ é ❞❡ ✈♦❝ês✱ q✉❡ ♣❛rt✐❝✐♣❛r❛♠ ❝♦♠ ♣❛♣❡❧ ♣r✐♥❝✐♣❛❧ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❡st❛ ❝♦♥str✉çã♦✳ ➚ ♠✐♥❤❛ ♠ã❡ ▲❡✐❞❡✱ ♣❡❧❛s ❧✐❣❛çõ❡s ❞✐ár✐❛s r❡♣❧❡t❛s ❞❡ ✉♠ ❛♠♦r ✐♥❝♦♥❞✐❝✐♦♥❛❧✱ ❧✐✈r❡ ❞❛s ❞✐stâ♥❝✐❛s✱ s✉♣❡r✐♦r ❛ t♦❞♦s ♦s ♣❡r❝❛❧ç♦s✳ ❆♦ ♠❡✉ ♣❛✐ ❆❧❡①❛♥❞r❡✱ ♣❡❧❛ s✉❛ ♣r❡♦❝✉♣❛çã♦ ❝♦♥st❛♥t❡ ❝♦♠ ♠❡✉ ❜❡♠ ❡st❛r q✉❡ ❢❡③ r♦♠♣❡r ♦s ❧✐♠✐t❡s ❞❡ ❞✐stâ♥❝✐❛ ❣❡♦❣rá✜❝❛ ♣❛r❛ ♠❡ ❞❛r ❛ ♠❡❧❤♦r ❡str✉t✉r❛ ❢ís✐❝❛ ❡ ❡♠♦❝✐♦♥❛❧ q✉❡ ♣♦❞❡r✐❛ t❡r✳ ➚ ♠✐♥❤❛ ✐r♠ã ❆❧❡①❛♥❞r❛✱ ♣❡❧❛ t♦r❝✐❞❛ ❡ ✐♥❝❡♥t✐✈♦ ❝♦♥st❛♥t❡s✳ ➚ ♠✐♥❤❛ ♥❛♠♦r❛❞❛ ❈❛r♦❧✱ ♣♦r s❡r ❛ ♠✐♥❤❛ ✐♥s♣✐r❛çã♦ ❡ ✈✐❞❛ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s q✉❡ t❡✈❡ ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ♠❡ ❛❥✉❞❛r✳ ❙✉❛ ♣r❡s❡♥ç❛ ❡♠ ♠✐♥❤❛ ✈✐❞❛✱ ❝♦r♦♦✉ ❝♦♠ ❛ ✈✐rt✉❞❡ ❞♦ ❛♠♦r ❛ ❢❡❧✐❝✐❞❛❞❡ ❞❡ r❡❛❧✐③❛r ❡ss❡ s♦♥❤♦✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❊③❡q✉✐❡❧ ❇❛r❜♦s❛✱ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❝♦♥✜❛♥ç❛ ❞❡♣♦s✐t❛❞❛ ❡♠ ♠✐♠✱
✐✐✐
❝♦♥st✐t✉í❞❛s ❞❡ ✉♠❛ ♠♦t✐✈❛çã♦ s❡♠♣r❡ ❝♦♥st❛♥t❡ ❡ ✉♠❛ ♦r✐❡♥t❛çã♦ té❝♥✐❝❛ ❞❡ ❣r❛♥❞❡③❛✳ ❈♦♠ ✈♦❝ê✱ ❛♣r❡♥❞✐ ❞❡ ❢♦r♠❛ ❧♦✉✈á✈❡❧✱ ♦ q✉❡ é s❡r ✉♠ ♣r♦✜ss✐♦♥❛❧ ❞❡ ❡①❝❡❧ê♥❝✐❛ ❡♠ ♠✐♥❤❛ ár❡❛✱ ♦♥❞❡ ♦ ❝r❡s❝✐♠❡♥t♦ ❛❝❛❞ê♠✐❝♦ ❡ ♣❡ss♦❛❧ ❡stã♦ ✐♥tr✐s❡❝❛♠❡♥t❡ ❧✐❣❛❞♦s✱ ♦♥❞❡ ♦ tr❛❜❛❧❤♦ sér✐♦ ❞❡✈❡ ❝❛♠✐♥❤❛r ✉♥✐❞♦ ❛ ❤✉♠✐❧❞❛❞❡ ❞❡ ❡♥s✐♥❛r ❡ ❝♦❧❛❜♦r❛r ❝♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♣❡ss♦❛❧ ♠út✉♦✳
➚ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❞❡st❛ t❡s❡✱ ♦s ♣r♦❢❡ss♦r❡s ▼❛r❝♦s P❡trú❝✐♦✱ ▼ár❝✐♦ ❇❛t✐st❛✱ ▼❛r❝♦s ▼♦♥t❡♥❡❣r♦✱ ❘♦❞♥❡② ❇✐❡③✉♥❡r ❡ ❊♠❡rs♦♥ ❆❜r❡✉✱ ♣❡❧❛ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ❡ ót✐♠❛s s✉❣❡stõ❡s q✉❡ ❡♥❣r❛❞❡❝❡♠ ♦ r❡s✉❧t❛❞♦ ✜♥❛❧ ❞❡st❛ t❡s❡✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛❣r❛❞❡ç♦ ❛ ▼❛r❝♦s P❡trú❝✐♦ ♣❡❧❛s ❡❞✐✜❝❛♥t❡s ❞✐s❝✉ssõ❡s ❛❝❡r❝❛ ❞❡ ♥♦♠❡♥❝❧❛t✉r❛s ❡♠ ♠✐♥❤❛ t❡s❡ ❡ ♦❜s❡r✈❛çõ❡s r❡❢❡r❡♥t❡s ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞✐s♣♦st❛ ♥❛ ❙❡çã♦ ✸✳✶✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋▼●✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ♣r♦❢❡ss♦r❡s ▼❛r❝❡❧♦ ❚❡rr❛✱ ▼❛r❝♦s ▼♦♥t❡♥❡❣r♦✱ ❊♠❡rs♦♥ ❆❜r❡✉✱ ❊③❡q✉✐❡❧ ❇❛r❜♦s❛✱ ❘♦❣ér✐♦ ▼♦❧ ❡ ▼ár✐♦ ❏♦r❣❡ ♣❡❧♦s ❝✉rs♦s ❛ ♥í✈❡❧ ❞❡ ❡①❝❡❧ê♥❝✐❛ ♠✐♥✐str❛❞♦s✱ ♠❡ ❞✐s♣♦♥❞♦ ❛ ✉♠❛ ❢♦r♠❛çã♦ ♣❡ss♦❛❧ só❧✐❞❛ ❝♦♠♦ ♣r♦❢❡ss♦r ❡ ♣❡sq✉✐s❛❞♦r✳ ❆❣r❛❞❡ç♦ ❛♦ Pr♦❢❡ss♦r ▼❛r✲ ❝♦s ▼♦♥t❡♥❡❣r♦ ♣❡❧❛ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡ ❡♠ r❡❝❡❜❡r✲♠❡ ❝♦♠♦ s❡✉ ❛❧✉♥♦ ♥♦ ✐♥í❝✐♦ ❞❡ ♠❡✉ ❞♦✉t♦r❛❞♦ ❡ ♣♦r s✉❛ ❝♦♥❞✉çã♦ ót✐♠❛ ❞♦ t❡♥s♦ ♣❡rí♦❞♦ ❞❡ q✉❛❧✐✜❝❛çã♦ ❞♦ ❞♦✉t♦r❛❞♦ ❛♦ ✐♥í❝✐♦ ❞❛ ♣r♦❞✉çã♦ ❞❡ ♠✐♥❤❛ t❡s❡✳
❆♦s ❛♠✐❣♦s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯❋▼●✱ ❛ q✉❡♠ t✐✈❡ ♣♦r ❢❛♠í❧✐❛ ♥❡ss❛s t❡rr❛s ❞✐st❛♥t❡s✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛♦ ❋❡❧✐♣❡ q✉❡ ♠❡ ❛❝♦♠♣❛♥❤♦✉ ✐♥ ❧♦❝♦ ❞❡s❞❡ ❛ ❝❤❡❣❛❞❛ ♥✉♠❛ ❝✐❞❛❞❡ ❞❡s❝♦♥❤❡❝✐❞❛ ❛té ❛ ✜♥❛❧✐③❛çã♦ ❞❡st❡ s♦♥❤♦✳ ❆♦s ❛♠✐❣♦s ❞❛ ❙❛❧❛ ✶✵✵✶✱ ❙á✈✐♦✱ ❘♦❞r✐❣♦ ❇♦t❡❧❤♦✱ ❊❧✐③❛✱ ❏❛✈✐❡r✱ ▲❡♦♥❛r❞♦✱ ❊❞✉❛r❞♦ ❈❛❜r❡r❛✱ ❈❛r❧♦s ❙❛❧❛③❛r✱ ●✐❧❜❡rt♦✱ ❊❞✐r✱ ▼♦❛❝✐r ❡ ❛ t♦❞♦s ❛❧✉♥♦s ❞❛ ♣ós ♣❡❧♦ ❛♠❜✐❡♥t❡ ❞❡ ❝♦♠♣❛♥❤❡✐r✐s♠♠♦✱ ❛♠✐③❛❞❡ ❡ ❛❧❡❣r✐❛ ❞✐s♣♦st♦s ❝♦♠♦ ❝r✉❝✐❛✐s ❛ ❡st❡ t❡♠♣♦ ❞❡ ❛♣r❡♥❞✐③❛❞♦ ❡♠ ♠✉✐t♦s s❡♥t✐❞♦s✳
◆ã♦ ♦❜st❛♥t❡ ❛ t♦❞♦ ❛♣♦✐♦ ❢❛♠✐❧✐❛r à ❞✐stâ♥❝✐❛ q✉❡ t✐✈❡✱ ❡♥❝♦♥tr❡✐ ❡♠ ❇❡❧♦ ❍♦r✐③♦♥t❡ ♠✉✐t♦s ❛♠✐❣♦s ❛ q✉❡♠ ❤♦❥❡ ♣♦ss♦ ❝♦♥s✐❞❡r❛r ❝♦♠♦ ❢❛♠í❧✐❛ ❞❡ ❜❡rç♦✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛❣r❛❞❡ç♦ à ❢❛♠í❧✐❛ ❞❡ ❙❡✉ ❊❞✉❛r❞♦ ❡ ❉♦♥❛ ❆♥❞r❡❛✱ ♠❡✉s ♣❛✐s ♠✐♥❡✐r♦s✱ q✉❡ ❞❡s❞❡ ♦ ♠♦♠❡♥t♦ q✉❡ ❝❤❡❣✉❡✐ ❡♠ s✉❛ ❝❛s❛ ♠❡ ❡♥s✐♥❛r❛♠ ♠✉✐t♦s ✈❛❧♦r❡s ❞❡ ❝♦♥✜❛♥ç❛✱ ❛♠✐③❛❞❡ ❡ ❛❥✉❞❛ ❛♦ ♣ró①✐♠♦✱ t❡st❛♠❡♥t♦s q✉❡ ❤♦❥❡ ❧❡✈♦ ♣❛r❛ ♠✐♥❤❛ ✈✐❞❛✳ ❆♦s ❛♠✐❣♦s ❞❛ ▲❡❣✐ã♦ ❞❡ ▼❛r✐❛✱ ♣♦r t♦❞♦ ♦ ❛♣♦✐♦ ❞❡ ❢é ❡ ♣❡ss♦❛❧ q✉❡ t✐✈❡✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ❉♦♥❛ ●❡♥✐✱ ♠❛✐s ✉♠❛ ♠ã❡ ♠✐♥❡✐r❛ q✉❡ ❣❛♥❤❡✐✳
✐✈
♠♦♠❡♥t♦s q✉❡ ♥❡❝❡ss✐t❡✐✳
❆♦ ❋✉♥❞♦ ❞❡ ❆♠♣❛r♦ à P❡sq✉✐s❛ ❡♠ ▼✐♥❛s ●❡r❛✐s✭❋❛♣❡♠✐❣✮✱ ❝✉❥♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ t♦r♥♦✉ ✈✐á✈❡❧ t♦❞♦ ♦ ♣❡r❝✉rs♦ ❞❡ ♠❡✉ ❞♦✉t♦r❛❞♦✱ ❥✉♥t♦ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ❝♦♥tr✐❜✉✐çõ❡s ❝✐❡♥tí✜❝❛s ❡ ♣❡ss♦❛✐s q✉❡ ♦❜t✐✈❡ ♥❡st❡s ❛♥♦s✳
❆ t♦❞♦s ♦s ❢❛♠✐❧✐❛r❡s ❡ ❛♠✐❣♦s ❞❡ ❆❧❛❣♦❛s ❡ P❡r♥❛♠❜✉❝♦ q✉❡✱ ♠❡s♠♦ ❞❡ ❧♦♥❣❡✱ t♦r❝❡✲ r❛♠ ❡ ♠❡ ❡♥✈✐❛r❛♠ ♠✉✐t❛s ♠❡♥s❛❣❡♥s ❞❡ â♥✐♠♦✳
❘❡s✉♠♦
◆❡st❛ t❡s❡✱ ♦❜t❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❞❡ r✐❣✐❞❡③ ❡♠ ✈❛r✐❡❞❛❞❡s q✉❡ s❛t✐❢❛③❡♠ ✉♠❛ ❡q✉❛çã♦ ❞♦ t✐♣♦ ❡stát✐❝❛ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛s ✈❛r✐❡❞❛❞❡s ❡stát✐❝❛s✱ ♦s ❘✐❝❝✐ ❙♦❧✐t♦♥s✱ ♦s ❙♦❧✐t♦♥s ❣❡♥❡r❛❧✐③❛❞♦s✱ ❛s ✈❛r✐❡❞❛❞❡s ❱✲❡stát✐❝❛s ❡ ❛s ✈❛r✐❡❞❛❞❡s ❊✐♥st❡✐♥ ❝♦♠ ✉♠❛
S1−❛çã♦ ❡stát✐❝❛✳ ❖s ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ♦❜t❡r ♥♦ss♦s ❞✐✈❡rs♦s r❡s✉❧t❛❞♦s sã♦ ❜❛s❡✲
❛❞♦s ♥❛ ❛♥á❧✐s❡ ❞❡ ✐❞❡♥t✐❞❛❞❡s ✐♥t❡❣r❛✐s ♥❛ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✱ t❛✐s ❝♦♠♦ ❛ ■❞❡♥t✐✲ ❞❛❞❡ ❞❡ P♦❤♦③✞❛❡✈✲❙❝❤ö❡♥ ❡ ❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❘❡✐❧❧②✱ ❡ ❡♠ té❝♥✐❝❛s ✈❛r✐❛❝✐♦♥❛✐s ✐♥s♣✐r❛❞❛s ♥❛ ❆♥á❧✐s❡ ●❡♦♠étr✐❝❛✳
P❛❧❛✈r❛s✲❈❤❛✈❡s✿ ❱❛r✐❡❞❛❞❡s ❊stát✐❝❛s❀ ❱❛r✐❡❞❛❞❡sV✲❡stát✐❝❛s❀ ■❞❡♥t✐❞❛❞❡ ❞❡ P♦❤♦③✞❛❡✈✲
❙❝❤ö❡♥❀ ❘✐❝❝✐ s♦❧✐t♦♥s✳
❆❜str❛❝t
■♥ t❤✐s t❤❡s✐s✱ ✇❡ ♦❜t❛✐♥ s♦♠❡ r❡s✉❧ts ♦❢ r✐❣✐❞✐t② ✐♥ ♠❛♥✐❢♦❧❞s ✇❤✐❝❤ s❛t✐s✜❡s ❛♥ ❡q✉❛✲ t✐♦♥ ♦❢ st❛t✐❝ t②♣❡✱ ❢♦r ❡①❛♠♣❧❡✱ t❤❡ st❛t✐❝ ♠❛♥✐❢♦❧❞s✱ t❤❡ ❘✐❝❝✐ ❙♦❧✐t♦♥s✱ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❙♦❧✐t♦♥s✱ t❤❡ ❱✲st❛t✐❝ ♠❛♥✐❢♦❧❞s ❛♥❞ t❤❡ ❊✐♥st❡✐♥ ♠❛♥✐❢♦❧❞s ✇✐t❤ ❛ S1−st❛t✐❝ ❛❝t✐♦♥✳ ❚❤❡
♠❡t❤♦❞s ✉s❡❞ ✐♥ ♦✉rs ✈❛r✐♦✉s r❡s✉❧ts ❛r❡ ❜❛s❡❞ ✐♥ t❤❡ ❛♥❛❧②s✐s ♦❢ ✐♥t❡❣r❛❧ ✐❞❡♥t✐t✐❡s ✐♥ ❘✐❡♠❛♥♥✐❛♥ ❣❡♦♠❡tr②✱ s✉❝❤ ❛s P♦❤♦③✞❛❡✈✲❙❝❤ö❡♥ ■❞❡♥t✐t② ❛♥❞ ❘❡✐❧❧② ■❞❡♥t✐t②✱ ❛♥❞ ✐♥ t❤❡ ✈❛r✐❛t✐♦♥❛❧ t❡❝❤♥✐q✉❡s ✐♥s♣✐r❡❞ ❜② t❤❡ ●❡♦♠❡tr✐❝ ❆♥❛❧②s✐s✳
❑❡②✲❲♦r❞s✿ ❙t❛t✐❝ ▼❛♥✐❢♦❧❞s❀ V✲st❛t✐❝ ♠❛♥✐❢♦❧❞s❀ P♦❤♦③✞❛❡✈✲❙❝❤ö❡♥ ■❞❡♥t✐t②❀ ❘✐❝❝✐
s♦❧✐t♦♥s✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ Pr❡❧✐♠✐♥❛r❡s ✾
✶✳✶ ◆♦t❛çõ❡s✱ t❡r♠✐♥♦❧♦❣✐❛ ❡ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✶ ❆ ❣❡♦♠❡tr✐❛ ✐♥trí♥s❡❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✷ ❆ ❣❡♦♠❡tr✐❛ ❡①trí♥s❡❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✳✸ ■❞❡♥t✐❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ ❱❛r✐❡❞❛❞❡s ❡stát✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸ ❱❛r✐❡❞❛❞❡s ❱✲❡stát✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✹ ▼étr✐❝❛s q✉❛s✐✲❊✐♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✷ ❆ ■❞❡♥t✐❞❛❞❡ ❞❡ P♦❤♦③✞❛❡✈✲❙❝❤ö❡♥ ❡ ❆♣❧✐❝❛çõ❡s ●❡♦♠étr✐❝❛s ✷✸
✷✳✶ ❯♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ P♦❤♦③✞❛❡✈✲❙❝❤ö❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷ ❆❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ❣❡♦♠étr✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷✳✶ ❙♦❧✐t♦♥s ❣❡♥❡r❛❧✐③❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷✳✷ ❖ ▲❡♠❛ ❞❡ ❆❧♠♦st✲❙❝❤✉r ❣❡♥❡r❛❧✐③❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✷✳✸ ❯♠ t❡♦r❡♠❛ ❞♦ t✐♣♦ ❆❧❡①❛♥❞r♦✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✸ ❆♣❧✐❝❛çõ❡s ❡♠ ✈❛r✐❡❞❛❞❡s ❡stát✐❝❛s ❡ V✲❡stát✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✸ ❉❡s✐❣✉❛❧❞❛❞❡s ❡ ❘❡s✉❧t❛❞♦s ❞❡ ❘✐❣✐❞❡③ ❡♠ ❱❛r✐❡❞❛❞❡s ❊stát✐❝❛s ❡ ❱✲
❡stát✐❝❛s ✺✶
✸✳✶ ❘❡s✉❧t❛❞♦s ❞❡ r✐❣✐❞❡③ s♦❜ ✉♠❛ ❤✐♣ót❡s❡ ❞❡ ♣✐♥❝❤✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✸✳✷ ❆ ■❞❡♥t✐❞❛❞❡ ❞❡ ❘❡✐❧❧② ❡ ❛♣❧✐❝❛çõ❡s ❡♠ ♠étr✐❝❛s V✲❡stát✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✸✳✷✳✶ ❯♠❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❘❡✐❧❧② ❣❡♥❡r❛❧✐③❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✸✳✷✳✷ ❯♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ✐♥t❡❣r❛❧ s♦❜r❡ ❛ ❢r♦♥t❡✐r❛ ❞❡ ✈❛r✐❡❞❛❞❡s ❱✲❡stát✐❝❛s ✻✵
✹ ❆❧❣✉♥s ❆✈❛♥ç♦s ❡♠ ❱❛r✐❡❞❛❞❡s ❊stát✐❝❛s ❞❡ ❉✐♠❡♥sã♦ ❆❧t❛ ✻✺ ✹✳✶ ❯♠ t❡♦r❡♠❛ ❞❡ ●✉rs❦② ✲ ❛♣❧✐❝❛çõ❡s ❡ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❣❛♣ ♥❛ ✹✲❡s❢❡r❛ ✳ ✳ ✳ ✻✽ ✹✳✷ ❙♦❜r❡ S1✲❛çõ❡s ❡stát✐❝❛s ❡♠ ✈❛r✐❡❞❛❞❡s ❊✐♥st❡✐♥ (n+ 1)✲❞✐♠❡♥s✐♦♥❛✐s ✳ ✳ ✳ ✼✶
✹✳✷✳✶ ❯♠ r❡s✉❧t❛❞♦ ❞❡ r✐❣✐❞❡③ ❡♠ ❞✐♠❡♥sã♦ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✹✳✷✳✷ ❘❡s✉❧t❛❞♦s ❞❡ r✐❣✐❞❡③ s♦❜ ✉♠❛ ❧✐♠✐t❛çã♦ ❞❛ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ♥♦
❜♦r❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✹✳✷✳✸ ❖ ❝❛s♦ ❞♦ ❜♦r❞♦ ❞❡s❝♦♥❡①♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵
■♥tr♦❞✉çã♦
❚❡♦r❡♠❛s ❞❡ r✐❣✐❞❡③ ❡♠ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛ s✉r❣❡♠ ❝♦♠♦ ✉♠❛ s❛❧✉t❛r ✐♥s♣✐r❛çã♦ ❞❡ ♣r♦♣❡♥s♦s ❡s❢♦rç♦s ♥❛ ár❡❛✱ ❜❡♠ ❝♦♠♦ ❛❜r❛❣❡♠ ❢❡rr❛♠❡♥t❛s ❞❛ ♠❛✐s ❞✐✈❡rs❛ ♥❛t✉r❡③❛✳ ❊①❡♠♣❧♦s ❞❡ ❜❡❧♦s t❡♦r❡♠❛s ♥❡st❡ ❡s❝♦♣♦ ♥❛ ❧✐t❡r❛t✉r❛ ♥ã♦ ♥♦s ❢❛❧t❛♠✱ ✐♥❞♦ ❞❡s❞❡ ♦s ❝❧ás✲ s✐❝♦s ❚❡♦r❡♠❛ ❞❡ ❇♦♥♥❡t✲▼②❡rs✱ ❚❡♦r❡♠❛ ❞❡ ❚♦♣♦♥♦❣♦✈ ❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❖❜❛t❛ às ♠✉✐t♦ r❡❝❡♥t❡s r❡s♦❧✉çõ❡s ❞❛ ❈♦♥❥❡❝t✉r❛ ❞❡ P♦✐♥❝❛ré ♣♦r P❡r❡❧♠❛♥✱ ❡ ❝♦♥❥❡❝t✉r❛ ❞❡ ❲✐❧❧♠♦r❡ ♣♦r ▼❛rq✉❡s✲◆❡✈❡s✱ ❛❧é♠ ❞♦ ❚❡♦r❡♠❛ ❞❛ ❊s❢❡r❛ ❉✐❢❡r❡♥❝✐á✈❡❧ ♣♦r ❇r❡♥❞❧❡✲❙❝❤ö❡♥✳ ❆❧é♠ ❞✐ss♦✱ té❝♥✐❝❛s ✉t✐❧✐tár✐❛s ❛ s✉❛s r❡s♦❧✉çõ❡s tr❛❞✉③❡♠ ❛ r✐q✉❡③❛ ❡ ❛❜r❛♥❣ê♥❝✐❛ ❞❡st❡s ❡s❢♦r✲ ç♦s✱ ♣❡r♣❛ss❛♥❞♦ ❞❛ t❡♦r✐❛ ❞❡ ❝♦♠♣❛r❛çã♦ ❛ ❛♥á❧✐s❡ ❞❡ ■❞❡♥t✐❞❛❞❡s ■♥t❡❣r❛✐s✱ ❝♦♠♦ ❛s ❞❡ ❘❡✐❧❧②✱ ▼✐♥❦♦✇s❦✐ ❡ ❑❛③❞❛♥✲❲❛r♥❡r✱ ❡ ✐♥❞♦ ❛♦ ❡♥❝♦♥tr♦ ❞❡ t❡♦r✐❛s ✈❛r✐❛❝✐♦♥❛✐s ❝♦♠♦ ♦s ✢✉①♦s ❣❡♦♠étr✐❝♦s ❞❡ ❘✐❝❝✐ ❡ ❞❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛✱ ❡ ❛ té❝♥✐❝❛ ❞♦ ♠✐♥✐✲♠❛①✳
◆❡st❡ s❡♥t✐❞♦✱ ✉♠ ♣r♦❜❧❡♠❛ q✉❡ ❡♥✈♦❧✈❡ r✐❣✐❞❡③ ❡ ♣❡r♠❡✐❛ ❛ ❤✐stór✐❛ r❡❝❡♥t❡ ❞❛ ❘❡❧❛✲ t✐✈✐❞❛❞❡ ●❡r❛❧ s❡ r❡❢❡r❡ à s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✐♥st❡✐♥ ❝♦♠♦ ✜♥s ❞❡ ♠♦❞❡❧❛❣❡♠ ❞❛ ❢♦r♠❛ ❞♦ ✉♥✐✈❡rs♦✳ ❖ ♦❜❥❡t♦ ❜❛s❡ ❞❡st❛ ár❡❛ ❞❡ ❡st✉❞♦ r❡s✐❞❡ ♥✉♠❛ (n+ 1)✲❞✐♠❡♥s✐♦♥❛❧
✈❛r✐❡❞❛❞❡ ▲♦r❡♥t③ (L, h) q✉❡ é ✉♠ ❡s♣❛ç♦ t❡♠♣♦ ❝♦♠ ❛ss✐♥❛t✉r❛(−,+,+, . . . ,+) s❛t✐s❢❛✲
③❡♥❞♦ ❛ ❡q✉❛çã♦ ❞❡ ❊✐♥st❡✐♥
Rich−
1
2Rhh+ Λh= 8πG
C4 Th,
♦♥❞❡ Rich é ♦ t❡♥s♦r ❞❡ ❘✐❝❝✐ ❞❡ h✱ Λ é ❛ ❝♦♥st❛♥t❡ ❝♦s♠♦❧ó❣✐❝❛✱ G é ❛ ❝♦♥st❛♥t❡ ❣r❛✈✐✲
t❛❝✐♦♥❛❧✱ C é ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ❡ Th é ♦ t❡♥s♦r ♠♦♠❡♥t♦✳ ◆❡st❡ tr❛❜❛❧❤♦✱ ✐♥✈♦❝❛r❡♠s
❛♦ ♣r♦❜❧❡♠❛ ♥♦ ✈á❝✉♦✱ ♦♥❞❡ Th = 0✳ ❖ ❡s♣❛ç♦ t❡♠♣♦ (L, h) é ❞✐t♦ ❡stát✐❝♦ s❡ ❡①✐st❡ ✉♠
❝❛♠♣♦ ❞❡ ❑✐❧❧✐♥❣ K ❞♦ t✐♣♦ t❡♠♣♦ ❡ ❣❧♦❜❛❧♠❡♥t❡ ❞❡✜♥✐❞♦✱ ❝✉❥❛ ❞✐str✐❜✉✐çã♦ ♦rt♦❣♦♥❛❧ é
✐♥t❡❣rá✈❡❧ ✭❝❢✳ ❙❡çã♦ ✸✳✹ ❞❡ ❬✹✸❪✮✳ ❆ ❡①✐stê♥❝✐❛ ❞❡ t❛❧ ❝❛♠♣♦ ❞❡ ❑✐❧❧✐♥❣ ❞❡t❡r♠✐♥❛ ✉♠❛ ❡str✉t✉r❛ ❞❡ ♣r♦❞✉t♦ ✇❛r♣❡❞ ❡♠ L❝♦♠♦
■♥tr♦❞✉çã♦ ✷
♦♥❞❡(Mn, g)é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❞❡ ❞✐♠❡♥sã♦n❝♦♠♣❛❝t❛✱ ❝♦♥❡①❛ ❡f :M
−→
Ré ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ ❞❡♥♦♠✐♥❛❞❛ ❞❡ ♣♦t❡♥❝✐❛❧ ❡stát✐❝♦✳ ❖ ❝❛♠♣♦ ∂
∂t s❡rá ♦ ❝❛♠♣♦ ❞❡ ❑✐❧✲
❧✐♥❣ ❡♠ q✉❡stã♦✳ ❈♦♠ ✉♠ ❝á❧❝✉❧♦ s✐♠♣❧❡s ❛ ♣r♦❞✉t♦s ✇❛r♣❡❞ ✭❝❢✳ ❬✶✵❪✱ Pr♦♣♦s✐çã♦ ✾✳✶✵✻✮✱ ❛ ❝♦♥❞✐çã♦ ❞❡ ❡st❛t✐❝✐❞❛❞❡ ❞❡ L é tr❛❞✉③✐❞❛ ♥♦ s❡❣✉✐♥t❡ Pr♦❜❧❡♠❛ ❙♦❜r❡❞❡t❡r♠✐♥❛❞♦ ❡♠ M
D2f −f Ricg+ Λf g = 0, ❡ ∆f =−Λg, ✭✷✮
♦♥❞❡ ❝♦♥s✐❞❡r❛♠♦sD2 ❡∆✱ ♦ ❍❡ss✐❛♥♦ ❡ ♦ ▲❛♣❧❛❝✐❛♥♦ ❛❞✈✐♥❞♦s ❞❛ ♠étr✐❝❛ g✳ ❉❡♥♦t❛♠♦s
(Mn, g, f) ❞❡ tr✐♣❧❛ ❡stát✐❝❛✳ ❯♠❛ ♦❜s❡r✈❛çã♦ ✐♠♣♦rt❛♥t❡ ❞❡st❛ ❢♦r♠✉❧❛çã♦ ❡❧í♣t✐❝❛ s❡
❞❡✈❡ ❛ ✈❡r✐✜❝❛çã♦ q✉❡ M t❡♠ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ❝♦♥st❛♥t❡ Rg = Λ(n−1)✳ ❆❧é♠ ❞✐ss♦✱
Σ = f−1(0) é ✉♠❛ ❤✐♣❡r❢í❝✐❡ ❢❡❝❤❛❞❛ ❡ ♠❡r❣✉❧❤❛❞❛ ❡♠ M✱ ♣♦ss✐✈❡❧♠❡♥t❡ ❝♦♠ ✈ár✐❛s
❝♦♠♣♦♥❡♥t❡s ❝♦♥❡①❛s ✭❝❢✳ ❡st❛s ❛✜r♠❛çõ❡s ♥♦ ❚❡♦r❡♠❛ ✶✳✶✮✳ ❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❢ís✐❝♦✱ ❛s ❤✐♣❡r❢í❝✐❡s R×Σ sã♦ ❝❤❛♠❛❞❛s ❞❡ ❤♦r✐③♦♥t❡s ❞❡ ❡✈❡♥t♦s ❡ t❡♠ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ❡st✉❞♦ ❞❡
❜✉r❛❝♦s ♥❡❣r♦s✭❝❢✳ ❬✹✸❪ ❡ ❬✹✺❪✱ ♣♦r ❡①❡♠♣❧♦✮✳
❖ ❡①❡♠♣❧♦ ♠♦❞❡❧♦ ❞❡ r✐❣✐❞❡③ ♣❛r❛ ✈❛r✐❡❞❛❞❡s ❡stát✐❝❛s ❝♦♠ ❝♦♥st❛♥t❡ ❝♦s♠♦❧ó❣✐❝❛
Λ >0 é ❞❛❞♦ ♣❡❧♦ ❤❡♠✐s❢ér✐♦ ❝❛♥ô♥✐❝♦ ❞❡ r❛✐♦r =pn
Λ✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r (S
n
+( pn
Λ), gst)✳
❈♦♥s✐❞❡r❛♥❞♦ ❛ ❢✉♥çã♦ ❛❧t✉r❛ h ♥❡st❛ ✈❛r✐❡❞❛❞❡ ♣♦❞❡✲s❡ ✈❡r✐✜❝❛r q✉❡ ❛ ♠❡s♠❛ s❛t✐s❢❛③
✭✷✮✳ ❉❡st❛ ❢♦r♠❛✱ ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡s♣❛ç♦✲t❡♠♣♦
(L+, h) = (R×Sn
+,−h2dt2+gst)
é ✉♠❛ s♦❧✉çã♦ ❞❛ ❊q✉❛çã♦ ❞❡ ❊✐♥st❡✐♥ ♥♦ ✈á❝✉♦✳ ❊st❡ é ♦ ❡①❡♠♣❧♦ ❞❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ✈❛r✐❡❞❛❞❡s✱ s♦❧✉çõ❡s ❡①❛t❛s ❞❛ ❊q✉❛çã♦ ❞❡ ❊✐♥st❡✐♥✱ ❝❤❛♠❛❞❛s ❞❡ ❡s♣❛ç♦s ❞❡ ❉❡ ❙✐tt❡r ✭❝❢✳ ❙❡çã♦ ✺✳✸ ❞❡ ❬✹✸❪✮✳ ❆❧é♠ ❞✐ss♦✱ é ✉♠ ❝♦♠♣♦♥❡♥t❡ ❞❡ r✐❣✐❞❡③ ♣❛r❛ ✉♠❛ ❢❛♠♦s❛ ❝♦♥❥❡❝t✉r❛ ❢♦r♠✉❧❛❞❛ ♣♦r ❇♦✉❝❤❡r✲●✐❜❜♦♥s✲❍♦r♦✇✐t③✭❬✶✹❪✮ ❡ ❛❝❧❛♠❛❞❛ ❝♦♠♦ ◆♦✲❍❛✐r ❈♦♥❥❡❝t✉r❡✿
❈♦♥❥❡❝t✉r❛ ✵✳✶ ✭◆♦✲❍❛✐r ❈♦♥❥❡❝t✉r❡✮✳ ❖ ú♥✐❝♦ ❡s♣❛ç♦✲t❡♠♣♦ ❡stát✐❝♦ ❝♦♠ ❝♦♥st❛♥t❡ ❝♦s♠♦❧ó❣✐❝❛ Λ > 0 ❡ ❤♦r✐③♦♥t❡ ❝♦s♠♦❧ó❣✐❝♦ ❞❡ ❡✈❡♥t♦s ❝♦♥❡①♦ é ♦ ❡s♣❛ç♦ ❞❡ ❉❡ ❙✐tt❡r
❞❡ r❛✐♦ pn
Λ✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❛ ú♥✐❝❛ tr✐♣❧❛ ❡stát✐❝❛ (M
n, g, f) ❝♦♠ ❜♦r❞♦ ❝♦♥❡①♦
❡ ❝♦♥st❛♥t❡ ❝♦s♠♦❧ó❣✐❝❛ Λ > 0 é ❞❛❞❛ ♣❡❧♦ ❤❡♠✐s❢ér✐♦ ❝❛♥ô♥✐❝♦ (Sn
+( pn
Λ), gst)✱ ♦♥❞❡ ♦
♣♦t❡♥❝✐❛❧ ❡stát✐❝♦ f é ❞❛❞♦ ♣❡❧❛ ❢✉♥çã♦ ❛❧t✉r❛✳
❙❛❧✐❡♥t❛♠♦s q✉❡ ❛ ❝♦♥❡①✐❞❛❞❡ ♥❛ ❝♦♥❥❡❝t✉r❛ ❛❝✐♠❛ é ❡ss❡♥❝✐❛❧ ✭❝❢✳ ❬✺✼❪✮✳ ❆ ♣❛rt✐r ❞❡st❡ ♠♦♠❡♥t♦ ❞❛ ❤✐stór✐❛✱ ✈ár✐♦s tr❛❜❛❧❤♦s ♥❛ ❞✐r❡çã♦ ❞❡ ❞❡♠♦♥str❛r ❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❡s♣❛ç♦ ❞❡ ❉❡ ❙✐tt❡r t❡♠ ❛❧♠❡❥❛❞♦ ❡s❢♦rç♦s ❝♦♠ ❛ ❛❞✐çã♦ ❞❡ ♥♦✈❛s ❤✐♣ót❡s❡s q✉❡ ❝♦rr♦❜♦r❛♠ ♣♦s✐t✐✈❛♠❡♥t❡ ♣❛r❛ ❛ ❛ss❡rt✐✈❛ ❞❛ ◆♦✲❍❛✐r ❈♦♥❥❡❝t✉r❡✳ ❇♦✉❝❤❡r✱ ●✐❜❜♦♥s ❡ ❍♦r♦✇✐t③ ✭❬✶✹❪✮ ♣r♦✈❛r❛♠ ✉♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ♣♦s✐t✐✈❛ ✭Λ>0✮✱ ♦♥❞❡ ✉♠❛
■♥tr♦❞✉çã♦ ✸
tr❛❜❛❧❤♦ ❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ✭❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦ ❛♥t✐✲❞❡ ❙✐tt❡r✮ ♣❛r❛ ♦ ❝❛s♦ ❞❡ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ♥❡❣❛t✐✈❛ ✭✐st♦ é Λ<0✮✳ ❇♦✉❝❤❡r ✭❬✶✷❪✮ ❡ ❋r✐❡❞r✐❝❤ ✭❬✸✷❪✮ ❞❡♠♦♥str❛r❛♠
♦ r❡s✉❧t❛❞♦ ❛sss✉♠✐♥❞♦ ✉♠❛ ❝♦♠♣❛❝t✐✜❝❛çã♦ ❞❡ P❡♥r♦s❡ ❞♦ ❡s♣❛ç♦ t❡♠♣♦ ✉♥✐❞♦ ❛ ❝❡rt❛s ❝♦♥❞✐çõ❡s ♥♦s ✜♥s ❝♦♥❢♦r♠❡s✳ ●❛❧❧♦✇❛②✭❬✸✹❪✮ ❞❡♠♦♥str♦✉ ♦ r❡s✉❧t❛❞♦ q✉❛♥❞♦ ♦ ❡s♣❛ç♦ t❡♠♣♦ ❝♦♥té♠ ✉♠❛ ♥✉❧❧ ❧✐♥❡✳ ❈❤rús❝✐❡❧ ✭❬✷✻❪✮ ❞❡♠♦♥str♦✉ ♦ r❡s✉❧t❛❞♦ ♥✉♠❛ ❞✐♠❡♥sã♦ q✉❛❧q✉❡r n≥3 ❝♦♠ ❛❧❣✉♠❛s ❝♦♥❞✐çõ❡s s♦❜r❡ ❛ ❢r♦♥t❡✐r❛✳
❈♦♠♦ ❧✐♠❜♦ à ♠♦t✐✈❛çã♦ ❢ís✐❝❛ ❞❛❞❛✱ ❛✐♥❞❛ t❡♠♦s ♠✉✐t♦s ❡♥t❡s ♥❛ ❣❡♦♠❡tr✐❛ q✉❡ s❛✲ t✐❢❛③❡♠ ✉♠❛ ❡q✉❛çã♦ s✐♠✐❧❛r à ✭✷✮ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♦s ❘✐❝❝✐ ❙♦❧✐t♦♥s✱ ♦s ❛❧♠♦st ❘✐❝❝✐ s♦❧✐t♦♥s✱ ❛s ✈❛r✐❡❞❛❞❡s q✉❛s✐✲❊✐♥st❡✐♥ ❡✱ ♠❛✐s r❡❝❡♥t❡♠❡♥t❡✱ ❛s ✈❛r✐❡❞❛❞❡ V✲❡stát✐❝❛s ✭❝♦♥✲
❢❡r✐r ❞❡s❝r✐çõ❡s ♥♦s ❝❛♣ít✉❧♦s ❞❡ ♥♦ss♦ tr❛❜❛❧❤♦✮✳ ❊♠ ❝♦♠✉♠✱ ❤á ✉♠ ✐♥t❡r❡ss❡ ❡♠ ❡st❛✲ ❜❡❧❡❝❡r ♠♦❞❡❧♦s ♣❛r❛ r✐❣✐❞❡③ ❡ ♥♦✈♦s r❡s✉❧t❛❞♦s ♥❡st❛ ❞✐r❡çã♦ s♦❜ ❝♦♥❞✐çõ❡s ♥❛ ❣❡♦♠❡tr✐❛ ✐♥trís❡❝❛ ♣❛r❛ ✐♥❞✐❝❛r t❛✐s ♠♦❞❡❧♦s✳
P❛r❛♠❡tr❛❞♦ ♣♦r t❛✐s ♠♦t✐✈❛çõ❡s✱ ♥♦ss❛ t❡s❡ s❡ ❝♦♠♣r♦♠❡t❡ ❛ ❞❛r ❝♦♥tr✐❜✉✐çõ❡s ❛ ✉♠❛ r❡❝❡♥t❡ ❣❛♠❛ ❞❡ tr❛❜❛❧❤♦s ❞❡ ❝❧❛ss✐✜❝❛çã♦ ❡ r✐❣✐❞❡③ ♥♦ tr❛t♦ ❞❡ ✈❛r✐❡❞❛❞❡s ❞♦ t✐♣♦ ❡stát✐❝❛ ❡ ❝♦rr❡❧❛t❛s✱ ✐♥❢❡r✐♥❞♦ té❝♥✐❝❛s q✉❡ r❡s✐❞❡♠✱ ❡s♣❡❝✐❛❧♠❡♥t❡✱ ♥❛ ✉t✐❧✐③❛çã♦ ❞❡ ■❞❡♥t✐❞❛❞❡s ❣❡♦♠étr✐❝❛s✱ ❝♦♠♦ ❛ ■❞❡♥t✐❞❛❞❡ ❞❡ P♦❤♦③✞❛❡✈✲❙❝❤ö❡♥ ❡ ❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❘❡✐❧❧②✱ à ✉t✐❧✐③❛çã♦ ❞❡ ♠ét♦❞♦s ❞❡ ❡①✐stê♥❝✐❛ ❞❡ s✉♣❡r❢í❝✐❡s ❡stá✈❡✐s✳ ◆♦ss❛ t❡s❡ é ♦r❣❛♥✐③❛❞❛ ❡♠ ✹ ❝❛♣ít✉❧♦s ❝✉❥♦ ❛✈❛♥ç♦ ❡♠ r❡❧❛çã♦ à ❧✐t❡r❛t✉r❛ ❡①✐st❡♥t❡ ❞❡s❝r❡✈❡♠♦s ❛ s❡❣✉✐r✳
❖ ❝❛♣ít✉❧♦ ✶ é ✐♥tr♦❞✉tór✐♦ às ♥♦t❛çõ❡s ❡ r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ❛ s❡r❡♠ ✉t✐❧✐③❛❞♦s ❛♦ ❞❡❝♦rr❡r ❞❡ ♥♦ss♦ t❡①t♦✳ ❆❧é♠ ❞✐ss♦✱ ✈✐s✐t❛♠♦s ❛s ❞❡✜♥✐çõ❡s ♣ré✈✐❛s ❞❡ ❛❧❣✉♥s ❡♥t❡s ❝✉❥❛s ♥♦tá✈❡✐s ♣r♦♣r✐❡❞❛❞❡s r❡❧❛t❛❞❛s s❡rã♦ ✐♥t❡❣r❛♥t❡s ❡♠ ♥♦ss♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ ❡♠ ❡s♣❡❝✐❛❧✱ ❛s ✈❛r✐❡❞❛❞❡s ❡stát✐❝❛s ❡ V✲❡stát✐❝❛s✳
❖ ❝❛♣ít✉❧♦ ✷ é ❞❡✈♦t❛❞♦ à ❛♣❧✐❝❛çã♦ ❞❛ ■❞❡♥t✐❞❛❞❡ ❞❡ P♦❤♦③✞❛❡✈✲❙❝❤♦❡♥ ❝♦♠ ✜♥s ❞❡ ♦❜t❡r ♥♦✈♦s r❡s✉❧t❛❞♦s ❞❡ r✐❣✐❞❡③✱ ❣❡♥❡r❛❧✐③❛r r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❡ ❞❛r ❞❡♠♦♥str❛çõ❡s ❝♦♥✲ s✐❞❡r❛✈❡❧♠❡♥t❡ ♠❛✐s s✐♠♣❧❡s ❞❛s ❞❡s❝r✐t❛s ♥❛ ❧✐t❡r❛t✉r❛ ❡①✐st❡♥t❡✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❝♦♥s✐❞❡r❛✲ ♠♦s ✉♠❛ ✈❡rsã♦ ❣❡♥❡r❛❧✐③❛❞❛ ❞❡st❛ ■❞❡♥t✐❞❛❞❡ ❝❧áss✐❝❛ ♣❛r❛ ❞❡✜♥✐r ❡♥t❡s ♠❛✐s ❣❡♥ér✐❝♦s✱ ❡ ♦❜t❡r ❞❡♠♦♥str❛çõ❡s q✉❡ ✉t✐❧✐③❛♠ ❞✐r❡t❛♠❡♥t❡ ❛ ✐❞❡♥t✐❞❛❞❡✳ P♦r ❡①❡♠♣❧♦✱ ❣❡♥❡r❛❧✐③❛♠♦s ❛ ♥♦çã♦ ❞❡ ❛❧♠♦st ❘✐❝❝✐ s♦❧✐t♦♥ ♣❛r❛ ♦❜t❡r ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❥✉♥t♦ ❛ ✉♠❛ ♣r♦✈❛ ❞✐st✐♥t❛ ❡ ❜❡♠ ♠❛✐s ❞✐r❡t❛ ❞♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✱ ❛♥✉♥❝✐❛❞♦ ♣r✐♠♦r❞✐❛❧♠❡♥t❡ ♣♦r ❇❛rr♦s ❡t✳ ❛❧ ❡♠ ❬✼❪ ❡ ❝♦♥♦s❝♦ ♦❜t✐❞♦ ❝♦♠♦ ❝♦r♦❧ár✐♦✳
❚❡♦r❡♠❛ ✵✳✶✳ ❈♦♥s✐❞❡r❡ (Mn, g, X, λ) ✱ n ≥ 3✱ ✉♠ ❛❧♠♦st ❘✐❝❝✐ s♦❧✐t♦♥ ♥ã♦ tr✐✈✐❛❧
❢❡❝❤❛❞♦ ❡ ♦r✐❡♥t❛❞♦ ❝♦♠ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ❝♦♥st❛♥t❡✳ ❊♥tã♦ (Mn, g) é ✐s♦♠étr✐❝♦ à ❡s❢❡r❛
❊✉❝❧✐❞❡❛♥❛ (Sn, g
st) ❡ (Mn, g, X, λ) é ✉♠ ❘✐❝❝✐ s♦❧✐t♦♥ ❣r❛❞✐❡♥t❡✳
■♥tr♦❞✉çã♦ ✹
r❛❧✐③❛çã♦ ❥✉♥t♦ ❛ ✉♠❛ ❞❡♠♦♥str❛çã♦ ♠❛✐s s✐♠♣❧❡s ❞❡ ✉♠ r❡s✉❧t❛❞♦ ♣❛r❛ h✲❛❧♠♦st ❘✐❝❝✐
s♦❧✐t♦♥s ♦❜t✐❞♦ ♣♦r ❳✐❛ ❡t✳ ❛❧ ❡♠ ❬✸✼❪✳
❚❛♠❜é♠ ❛♣❧✐❝❛r❡♠♦s ❛ ■❞❡♥t✐❞❛❞❡ ❞❡ P♦❤♦③✞❛❡✈✲❙❝❤♦❡♥ ❣❡♥❡r❛❧✐③❛❞❛ ♣❛r❛ ♦❜t❡r ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♥❛ ❞✐r❡çã♦ ❞❡ ❣❡♥❡r❛❧✐③❛r ♦ ▲❡♠❛ ❞❡ ❆❧♠♦st✲❙❝❤✉r ♣❛r❛ t❡♥s♦r❡s ♠❛✐s ❣❡♥ér✐❝♦s ❡ ✈❛r✐❡❞❛❞❡s ❝♦♠ ❢r♦♥t❡✐r❛✳ ❆❧é♠ ❞✐ss♦✱ ♦❜t❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ q✉❡ t♦♠❛ ❛ ❞✐r❡çã♦ ❞❡ ✉♠ t❡♦r❡♠❛ ❞♦ t✐♣♦ ❆❧❡①❛♥❞r♦✈ ♣❛r❛ ❤✐♣❡r❢í❝✐❡s ✐♠❡rs❛s✳
❚❡♦r❡♠❛ ✵✳✷✳ ❈♦♥s✐❞❡r❡ (Σn, g
Σ)✉♠❛ ❤✐♣❡r❢í❝✐❡ ❢❡❝❤❛❞❛ ❡ ✐♠❡rs❛ ♥✉♠❛ ✈❛r✐❡❞❛❞❡ ❊✐♥s✲
t❡✐♥ (Mn+1, g)✱ ❝♦♠ gΣ ❛ s✉❛ ♠étr✐❝❛ ✐♥❞✉③✐❞❛✳ ❙✉♣♦♥❤❛ q✉❡ ❡①✐st❡♠ ❢✉♥çõ❡s s✉❛✈❡s λ : Σn →R ❡ f : Σn →R s❛t✐s❢❛③❡♥❞♦
II+D2
Σf =λgΣ, ✭✸✮
♦♥❞❡ II é ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ Σ✳ ❙❡ ❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❞❡ Σ é ❝♦♥st❛♥t❡✱
❡♥tã♦ Σ é ✉♠❛ ❤✐♣❡r❢í❝✐❡ t♦t❛❧♠❡♥t❡ ✉♠❜í❧✐❝❛✳
❯♠❛ s❡çã♦ ❞♦ ❝❛♣ít✉❧♦ ✷ é ❞❡❞✐❝❛❞❛ ❛ ♠♦str❛r ❛ ❡✜❝á❝✐❛ ♥♦ tr❛t♦ ❞❛ ■❞❡♥t✐❞❛❞❡ ❞❡ P♦❤♦③✞❛❡✈✲❙❝❤♦❡♥ ❛♣❧✐❝❛❞❛ ❛s ✈❛r✐❡❞❛❞❡s V✲❡stát✐❝❛s✱ ❛s q✉❛✐s ♣♦❞❡♠ s❡r ✈✐st❛s ❝♦♠♦ ✉♠❛
❣❡♥❡r❛❧✐③❛çã♦ ♥❛t✉r❛❧ ❞❛s ✈❛r✐❡❞❛❞❡s ❡stát✐❝❛s✳ ❉❡st❛ ❢♦r♠❛✱ ❣❡♥❡r❛❧✐③❛r❡♠♦s ❛ ❡st❛ ❝❧❛ss❡ ❞❡ ✈❛r✐❡❞❛❞❡s ❛❧❣✉♥s ❝é❧❡❜r❡s r❡s✉❧t❛❞♦s ❛❜♦r❞❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ ✈❛r✐❡❞❛❞❡s ❡stát✐❝❛s✱ ❝❛♠✐♥❤❛♥❞♦ ♥❛ ❞✐r❡çã♦ ❞❡ ✉♠❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞♦ t✐♣♦ ❇♦✉❝❤❡r✲●✐❜❜♦♥s✲❍♦r♦✇✐t③✭❬✶✹❪✮ ♣❛r❛ V✲❡stát✐❝❛s✿
❚❡♦r❡♠❛ ✵✳✸✳ ❈♦♥s✐❞❡r❡ (Mn, g)✉♠❛ ✈❛r✐❡❞❛❞❡ ❝♦♠♣❛❝t❛✱ V✲❡stát✐❝❛ ♣♦s✐t✐✈❛ ❝♦♠ ❢r♦♥✲
t❡✐r❛ ∂M =
l
S
α=1
Σα✱ ❡ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r Rg =εn(n−1)✱ ♦♥❞❡ ε =−1,0,1✳ ❈♦♥s✐❞❡r❡ ❛s
❝♦♥st❛♥t❡s κi =|∇λ||Σi✳ ❊♥tã♦✱ ❛ s❡❣✉✐♥t❡ ❞❡s✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡
l
X
α=1 Z
Σα
κα
Rgα−ε(n−2)(n−1)−
n−2
n−1H
2
dVΣα ≥0.
❆❧é♠ ❞♦ ♠❛✐s✱ ❛ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ (Mn, g) é ✐s♦♠étr✐❝❛ ❛ ✉♠❛ ❜♦❧❛
❣❡♦❞és✐❝❛ ❡♠ ✉♠ ❞♦s ❡s♣❛ç♦s Rn✱ Sn ♦✉ Hn✳
❚❡♦r❡♠❛ ✵✳✹✳ ❈♦♥s✐❞❡r❡ (M3, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ V✲❡stát✐❝❛ ♣♦s✐t✐✈❛ ❝♦♠
❢r♦♥t❡✐r❛ ❝♦♥❡①❛ ❡ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r Rg = 6ε✱ ♦♥❞❡ ε = −1,0,1✳ ❙❡ ε = −1✱ ❛ss✉♠❛
❛❞✐❝✐♦♥❛❧♠❡♥t❡ q✉❡ H ≥2✳ ❊♥tã♦
4π≥
ε+1 4H
2
■♥tr♦❞✉çã♦ ✺
❆❧é♠ ❞✐ss♦✱ ❛ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ (M3, g)é ✐s♦♠étr✐❝❛ ❛ ✉♠❛ ❜♦❧❛ ❣❡♦❞és✐❝❛
❡♠ ✉♠ ❞♦s ❡s♣❛ç♦s R3✱ S3 ♦✉ H3✳
P♦r ✜♠✱ ♥❡st❡ ❝❛♣ít✉❧♦ ♦❜t❡♠♦s ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡ ✉♠ r❡s✉❧t❛❞♦ ❛❝❧❛♠❛❞♦ ♣♦r ▼✐❛♦ ❡ ❚❛♠✭❬✻✹❪✮✱ ❥✉♥t♦ ❛ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞✐st✐♥t❛ q✉❡ ♥♦✈❛♠❡♥t❡ ✉t✐❧✐③❛ ❛ ■❞❡♥t✐❞❛❞❡ ❞❡ P♦❤♦③✞❛❡✈✲❙❝❤ö❡♥✳ ❉❡ ❢❛t♦✱ s✉❛ ✉❧t✐❧✐③❛çã♦ ✐rá s❡ ♠♦str❛r ✉♠ ❝❛t❛❧✐s❛❞♦r à s✉❛ r❡s♦❧✉çã♦ ❥✉♥t♦ ❛ ✉♠❛ s✉❜tr❛çã♦ ❞❛s ❤✐♣ót❡s❡s ❞♦ ♣r♦❜❧❡♠❛ ✐♥✐❝✐❛❧♠❡♥t❡ ♣r♦♣♦st♦✳ ❙❛❧✐❡♥t❛♠♦s q✉❡ ♦ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ ♣♦r ❬✻✹❪ ❞✐❢❡r❡ ❞♦ ❞❡s❝r✐t♦ ❛❜❛✐①♦ ♣♦r s✉❛ r❡str✐çã♦ à ❞✐♠❡♥sã♦ ❞❛ ✈❛r✐❡❞❛❞❡ ❡ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ ❛ ❝✉r✈❛t✉r❛ é ♥ã♦ ♣♦s✐t✐✈❛✳ ❚❛✐s ❤✐♣ót❡s❡s sã♦ ❡①♣❧✐❝❛❞❛s ♣❡❧❛ ❢♦r♠❛ ❞✐st✐♥t❛ ❞❛ ♥♦ss❛ ❡♠ tr❛t❛r ♦ ♠❡s♠♦ ♣r♦❜❧❡♠❛✱ ❥á q✉❡ ♦ r❡s✉❧t❛❞♦ ❞❡❧❡s ✐♥❢❡r❡ ❛ té❝♥✐❝❛s ❛❞✈✐♥❞❛s ❛♦ ❚❡♦r❡♠❛ ❞❛ ▼❛ss❛ P♦s✐t✐✈❛✳
❚❡♦r❡♠❛ ✵✳✺✳ ❈♦♥s✐❞❡r❡ (Mn, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛ ❞❡ ❞✐♠❡♥sã♦ n
❝♦♠ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ❝♦♥st❛♥t❡ Rg = εn(n−1)✳ ❙✉♣♦♥❤❛ q✉❡ M ♣♦ss✉✐ ✉♠❛ ❢r♦♥t❡✐r❛
❝♦♥❡①❛ ❡ s✉❛✈❡ Σ✱ t❛❧ q✉❡ g é ✉♠ ♣♦♥t♦ ❝rít✐❝♦ ❞♦ ❢✉♥❝✐♦♥❛❧ ✈♦❧✉♠❡ ❡♠ MK
γ✱ K =
εn(n−1)✳ ❙✉♣♦♥❤❛ q✉❡ (Σ, γ) é ✐s♦♠étr✐❝❛ ❛ ✉♠❛ ❤✐♣❡r❢í❝✐❡ t♦t❛❧♠❡♥t❡ ✉♠❜í❧✐❝❛ Σε ❡♠
Eε✭♦♥❞❡ Eε é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❊✐♥st❡✐♥ ❝♦♠ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r εn(n−1) ❡ ε∈ {−1,0,1}✮✳
❊♥tã♦✱
Hε≥H ,
♦♥❞❡ H ❡ Hε sã♦ ❛s ❝✉r✈❛t✉r❛s ♠é❞✐❛s ❞❡ Σ ❡ Σε✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡
s❡✱ ❡ s♦♠❡♥t❡ s❡✱ (M, g) é ✐s♦♠étr✐❝❛ ❛ ✉♠ ❜♦❧❛ ❣❡♦❞és✐❝❛ ❡♠ Rn✱ Sn ♦✉ Hn✳
❖ ❝❛♣ít✉❧♦ ✸✱ t❡♠ ❛ s❡✉ ✐♥í❝✐♦ ❛❧❣✉♥s t❡♦r❡♠❛s ❞❡ r✐❣✐❞❡③ ♣❛r❛ ✈❛r✐❡❞❛❞❡s ❡stát✐❝❛s ❡
V✲❡stát✐❝❛s ♦❜t✐❞♦s ♣♦r ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ ♣✐♥❝❤✐♥❣✳ ❆ ♦❜s❡r✈❛çã♦ ❝r✉❝✐❛❧ à ♣r✐♠❡✐r❛ s❡çã♦
❞❡st❡ ❝❛♣ít✉❧♦ s❡ ❞❡✈❡ ❛ ♥♦t❛r q✉❡ ❛tr❛✈és ❞❡ ✉♠❛ ❝♦♠♣✉t❛çã♦ ❛❧❣é❜r✐❝❛ é ♣♦ssí✈❡❧ r❡❞✉③✐r ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ ♣✐♥❝❤✐♥❣ à ❝♦♥❞✐çã♦ ❞❡ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦ ♥❡❣❛t✐✈❛✳ ❆ ♣❛rt✐r ❞✐ss♦✱ ✉t✐❧✐③❛r❡♠♦s r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s ♣❛r❛ ✈❛r✐❡❞❛❞❡s ❝♦♠ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦ ♥❡❣❛t✐✈❛ ♣❛r❛ ❞❡♠♦♥str❛r ♥♦✈♦s r❡s✉❧t❛❞♦s ❡ ✐♥❢❡r✐r ♣r♦✈❛s ♠❛✐s s✐♠♣❧❡s ❞❡ r❡s✉❧t❛❞♦s ❥á ❛❝❧❛♠❛❞♦s✳ P♦r ❡①❡♠♣❧♦✱ ❡♠ ❬✸❪ ▲✳ ❆♠❜r♦③✐♦ ♦❜t❡♠ ✉♠ r❡s✉❧t❛❞♦ ❞❡ r✐❣✐❞❡③ ♣❛r❛ ✈❛r✐❡❞❛❞❡s ❡stát✐❝❛s tr✐❞✐♠❡♥s✐♦♥❛✐s s♦❜ ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ ♣✐♥❝❤✐♥❣ ❢❛③❡♥❞♦ ♦ ✉s♦ ❞❡ ✉♠❛ r♦❜✉st❛ ❋ór♠✉❧❛ ❞♦ t✐♣♦ ❇ö❝❤♥❡r ❡♠ ✉♠❛ ❞✐r❡çã♦ té❝♥✐❝❛✳ ❉❡ ✉♠❛ ♠❛♥❡✐r❛ ❝♦♥s✐❞❡r❛✈❡❧♠❡♥t❡ ♠❛✐s s✐♠♣❧❡s✱ ✉t✐❧✐③❛♥❞♦ ❛ ❣ê♥❡s❡ ❞❛ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦ ♥❡❣❛t✐✈❛✱ ❡st❡♥❞❡♠♦s ❡st❡ r❡s✉❧t❛❞♦✿
❚❡♦r❡♠❛ ✵✳✻✳ ❈♦♥s✐❞❡r❡ (M3, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ♦r✐❡♥tá✈❡❧✱ ❝♦♠♣❛❝t❛ ❡
❡stát✐❝❛ ❝♦♠ ❢r♦♥t❡✐r❛ Σ6=∅✳ ❆ss✉♠❛ q✉❡ Rg = 6 ❡
|Ric◦ g|2 ≤
R2
g
■♥tr♦❞✉çã♦ ✻
❊♥tã♦M ♣♦ss✉✐ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦ ♥❡❣❛t✐✈❛ ❡ ✉♠❛ ❞❛s ♠✉t✉❛♠❡♥t❡ ❡①❝❧✉s✐✈❛s ♦♣çõ❡s
❞❡✈❡ ♦❝♦rr❡r✿
✭✐✮ (M3, g) é ✐s♦♠étr✐❝❛ ❛♦ ❤❡♠✐s❢ér✐♦ ✉♥✐tár✐♦ ❝♦♠ ❛ ♠étr✐❝❛ ♣❛❞rã♦ (S3 +, gst)✳
✭✐✐✮ (M3, g) é ✐s♦♠étr✐❝❛ ❛♦ ❝✐❧✐♥❞r♦ (S2(√1
3)×[0,
π
√
3]), g0)✱ ♦♥❞❡ g0 é ❛ ♠étr✐❝❛ ♣r♦❞✉t♦✳
❆❧é♠ ❞✐ss♦✱ ♦ ♥♦ss♦ ♠ét♦❞♦ ♣r♦♣õ❡ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡st❡ t❡♦r❡♠❛ ♣❛r❛ ❞✐♠❡♥sõ❡s ❛❧t❛s✿
❚❡♦r❡♠❛ ✵✳✼✳ ❈♦♥s✐❞❡r❡ (Mn, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❡stát✐❝❛ ❝♦♠ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ♣♦s✐t✐✈❛✳
❙❡
|Ric◦ g|2 ≤
R2g
n(n−1)
❡♥tã♦ (Mn, g) t❡♠ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦ ♥❡❣❛t✐✈❛✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s
♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡✈❡♠ ♦❝♦rr❡r✿
✭✐✮ ❆ ❛♣❧✐❝❛çã♦ i∗ : Π(∂M)→Π(M) é s♦❜r❡❥❡t♦r❛✳
✭✐✐✮ ❆ ✈❛r✐❡❞❛❞❡ (Mn, g) é ✐s♦♠étr✐❝❛ ❛♦ q✉♦❝✐❡♥t❡ ❞❡ ✉♠ ❝✐❧✐♥❞r♦✳
❚❛♠❜é♠ ♦❜t❡♠♦s t❡♦r❡♠❛s ❞♦ ♠❡s♠♦ ❝❛❧✐❜r❡ ♣❛r❛ ✈❛r✐❡❞❛❞❡sV✲❡stát✐❝❛s✿
❚❡♦r❡♠❛ ✵✳✽✳ ❈♦♥s✐❞❡r❡ (M3, g) ✉♠❛ ✈❛r✐❡❞❛❞❡V✲❡stát✐❝❛ ♣♦s✐t✐✈❛ ❝♦♠ ❝✉r✈❛t✉r❛ ❡s❝❛✲
❧❛r ♣♦s✐t✐✈❛✳ ❙❡
|Ric◦ g|2 ≤
R2
g
6
❡♥tã♦ (M3, g) é ❞✐❢❡♦♠♦r❢❛ ❛ ✉♠❛ ❜♦❧❛ ❣❡♦❞és✐❝❛ ❡♠ S3✳
❚❡♦r❡♠❛ ✵✳✾✳ ❈♦♥s✐❞❡r❡ (Mn, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ V✲❡stát✐❝❛ ♣♦s✐t✐✈❛ ❝♦♠
❝✉r✈❛t✉r❛ ❡s❝❛❧❛r ♥ã♦ ♥❡❣❛t✐✈❛✳ ❙❡
|Ric◦ g|2 ≤
R2
g
n(n−1)
❡♥tã♦ (Mn, g) t❡♠ ❝✉r✈❛t✉r❛ ❞❡ ❘✐❝❝✐ ♥ã♦ ♥❡❣❛t✐✈❛✳ ❊♠ ♣❛rt✐❝✉❧❛r ∂M é ❝♦♥❡①❛✳ ❙❡ ∂M
é s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①❛✱ ❡♥tã♦ M é s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①❛✳
❆✐♥❞❛ ♥♦ ❝❛♣ít✉❧♦ ✸✱ ❢❛r❡♠♦s ♦ ✉s♦ ❞❡ ♦✉tr❛ ✐♠♣♦rt❛♥t❡ ✐❞❡♥t✐❞❛❞❡ ✐♥t❡❣r❛❧ ❞❛ ❣❡♦✲ ♠❡tr✐❛✱ ❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❘❡✐❧❧②✱ ❝♦♠ ✜♥s ❞❡ ♦❜t❡r ❡①t❡sõ❡s ❞❡ r❡s✉❧t❛❞♦s r❡❝❡♥t❡s ✐♥❢❡r✐❞♦s ❛❝❡r❝❛ ❞❡ ❞❡s✐❣✉❛❧❞❛❞❡s s♦❜r❡ ❛ ❢r♦♥t❡✐r❛ ❞❡ ✈❛r✐❡❞❛❞❡s ❡stát✐❝❛s ✭✈✐❞❡ ❬✺✺❪✱ ❬✼✻❪ ❡ ❬✻✶❪✮✳ ❉❡st❛ ❢♦r♠❛✱ tr❛t❛r❡♠♦s ♦ ❝❛s♦ V✲❡stát✐❝♦ ♣❛r❛ ♦❜t❡r r❡s✉❧t❛❞♦s ❝♦rr❡❧❛t♦s✱ ❝♦♠♦ ♣♦r
■♥tr♦❞✉çã♦ ✼
❚❡♦r❡♠❛ ✵✳✶✵✳ ❈♦♥s✐❞❡r❡ (M, g) ✉♠❛ ♥✲✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛ ❡ V✲❡stát✐❝❛
❝♦♠ ❢r♦♥t❡✐r❛ Σ ❡ ♣♦t❡♥❝✐❛❧ V✲❡stát✐❝♦ ♣♦s✐t✐✈♦ λ✳ ❈♦♥s✐❞❡r❡H ❡ II ❝♦♠♦ s❡♥❞♦ ❛ ❝✉r✈❛✲
t✉r❛ ♠❡❞✐❛ ❡ ❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ Σ ❡♠ (M, g)✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡ H > 0
❡ k ≤ 0 é ✉♠❛ ❝♦♥st❛♥t❡ ♥ã♦ ♣♦s✐t✐✈❛ t❛❧ q✉❡ Ric ≥ (n−1)kg✱ ❡♥tã♦ t❡♠♦s ❛ s❡❣✉✐♥t❡
❞❡s✐❣✉❛❧❞❛❞❡ ✐♥t❡❣r❛❧ s♦❜r❡ Σ
Z
Σ λ
[∆Ση+ (n−1)kη2]
H −II(∇Ση,∇Ση)
dσ ≥
Z
Σ
λ,ν[|∇Ση|2−(n−1)kη2]dσ, ✭✹✮
♣❛r❛ q✉❛❧q✉❡r ❢✉♥çã♦ η ❡♠ Σ✳ ❆❧é♠ ❞✐ss♦✱ ❛ ✐❣✉❛❧❞❛❞❡ ❡♠ ✭✸✳✻✮ ♦❝♦rr❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱
♦❝♦rr❡ ✉♠❛ ❞❛s ♦♣çõ❡s
✭✐✮ k = 0 ❡ η é ❛ r❡str✐çã♦ ❛ Σ ❞❡ ✉♠❛ ❢✉♥çã♦ u ❡♠ M s❛t✐s❢❛③❡♥❞♦ q✉❡ D2u= 0. ♦✉
✭✐✐✮ k < 0 ❡ g é ✉♠❛ ♠étr✐❝❛ ❊✐♥st❡✐♥ ❝♦♠ Ric = (n−1)kg✱ ❡ η é ❛ r❡str✐çã♦ ❛ Σ ❞❡
✉♠❛ ❢✉♥çã♦ u ❞❡✜♥✐❞❛ ❡♠ M s❛t✐s❢❛③❡♥❞♦ q✉❡ D2u+kug= 0✳
❖ ❝❛♣ít✉❧♦ ✹ t❡♠ ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ✐♥❢❡r✐r ♥♦✈♦s r❡s✉❧t❛❞♦s r❡❧❛❝✐♦♥❛❞♦s ❛ ✈❛r✐✲ ❡❞❛❞❡s ❡stát✐❝❛s ❞❡ ❞✐♠❡♥sã♦ ❛❧t❛✳ ❆ ♣❛rt✐r ❞❛ ❞❡s❝r✐çã♦ ♣✐♦♥❡✐r❛ ❞❡ ❙❡s❤❛❞r✐ ❬✽✷❪ ❛❝❡r❝❛ ❞❡ ✈❛r✐❡❞❛❞❡ ❊✐♥st❡✐♥ ♠✉♥✐❞❛s ❞❡ ✉♠❛ S1✲❛çã♦ ✐s♦♠étr✐❝❛ ❡stát✐❝❛✱ q✉❡ ❛ ✉♠ ❝❡rt♦ ♠♦❞♦
♣❛rt✐❝✉❧❛r✐③❛ ❛ ❞❡s❝r✐çã♦ ❞❡ ✈❛r✐❡❞❛❞❡ ❡stát✐❝❛ ❛❜♦r❞❛❞❛ ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s✱ ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ❛❧❣✉♥s t❡♦r❡♠❛s ❞❡ r✐❣✐❞❡③ q✉❛♥❞♦ t❡♠♦s ❡♠ ♠ã♦s ✉♠❛ ❤✐♣ót❡s❡ s♦❜ ❛ ❝✉r✈❛✲ t✉r❛ ❡s❝❛❧❛r ❞♦ ❜♦r❞♦ ❞❛ ✈❛r✐❡❞❛❞❡ ❡stát✐❝❛✳ P❛rt✐❝✉❧❛r♠❡♥t❡✱ ❙❡s❤❛❞r✐ ❡♠♣r❡❣❛ ❡♠ s❡✉ tr❛❜❛❧❤♦ ♦ ❝❛s♦ ❞❡ ✈❛r✐❡❞❛❞❡s ❡stát✐❝❛s tr✐❞✐♠❡♥s✐♦♥❛✐s✳
Pr❡❧✐♠✐♥❛r♠❡♥t❡✱ ❛tr❛✈és ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ❚❡♦r❡♠❛ ❞❡ ●✉rs❦② ✭❝❢✳ ❚❡♦r❡♠❛ ✹✳✷✮✱ ❝♦♥s❡❣✉✐♠♦s ✉♠❛ ❝♦♥s✐❞❡rá✈❡❧ ♠❡❧❤♦r❛ ❛ ✉♠ ❚❡♦r❡♠❛ ❞❡ ❍✳ ❙❡s❤❛❞r✐ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❞✐♠❡♥sã♦ n = 3✿
❚❡♦r❡♠❛ ✵✳✶✶✳ ❈♦♥s✐❞❡r❡ (N4, h) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❊✐♥st❡✐♥ ❡stát✐❝❛✱ ❢❡❝❤❛❞❛ ❡ ♥♦r♠❛✲
❧✐③❛❞❛ ❞❡ ❢♦r♠❛ q✉❡ Rich = 3h✳ ❈♦♥s✐❞❡r❡ (M3, g) ❛ ✈❛r✐❡❞❛❞❡ ❡stát✐❝❛ ❛ss♦❝✐❛❞❛✭❝❢✳
❉❡✜♥✐çã♦ ✹✳✶✮✳ ❙❡ K ≤ 3 ❡♠ ∂M✱ ❡♥tã♦ (N4, h) é ✐s♦♠étr✐❝❛ ❛ (S4, g0)✭❝❛s♦ ∂M s❡❥❛
❝♦♥❡①❛✮ ♦✉ (N4, h)é ✐s♦♠étr✐❝❛ ❛ (S2×S2, g
st)✭❝❛s♦ ∂M t❡♥❤❛ ♠❛✐s q✉❡ ✉♠❛ ❝♦♠♣♦♥❡♥t❡
❝♦♥❡①❛✮✳
■♥tr♦❞✉çã♦ ✽
❚❡♦r❡♠❛ ✵✳✶✷✳ ❈♦♥s✐❞❡r❡ (N, h) ✉♠❛(n+ 1)✲✈❛r✐❡❞❛❞❡ ❢❡❝❤❛❞❛✱ s✐♠♣❧❡s♠❡♥t❡ ❝♦♥❡①❛ ❡
❊✐♥st❡✐♥ ❡stát✐❝❛ ❝♦♠ ✈❛r✐❡❞❛❞❡ ❡stát✐❝❛ ❛ss♦❝✐❛❞❛ (M, g)✳ ❙❡ Rg = (n−1)(n−2) ♣❛r❛
t♦❞♦s ♦s ♣♦♥t♦s ❞❡ ∂M✱ ❡♥tã♦ (N, h) é ✐s♦♠étr✐❝❛ ❛ (Sn+1, gSn+1)✳
❱❡r❡♠♦s q✉❡ ♦ ❝❛s♦ ❞♦ ❜♦r❞♦ ❞❡s❝♦♥❡①♦ ♥♦s ♣r♦♣õ❡ ✉♠❛ ❛❜♦r❞❛❣❡♠ ❜❡♠ ❞✐st✐♥t❛ ❞♦ ❝❛s♦ ❛♥t❡r✐♦r✱ q✉❡ r❡s✐❞❡ ❢♦rt❡♠❡♥t❡ ♥❛ t❡♦r✐❛ ❞❡ ❡①✐stê♥❝✐❛ ❞❡ ❤✐♣❡r❢í❝✐❡s ♠í♥✐♠❛s ❡stá✈❡✐s ❡ ♥♦s ❛❝❛rr❡t❛ ✉♠❛ ❧✐♠✐t❛çã♦ à ❞✐♠❡♥sã♦✳ ❉❡ ❢❛t♦✱ ❝♦♥s❡❣✉✐♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
❚❡♦r❡♠❛ ✵✳✶✸✳ ❈♦♥s✐❞❡r❡Nn+1 ✉♠❛ ✈❛r✐❡❞❛❞❡ ❊✐♥st❡✐♥ ❡stát✐❝❛✱ ❢❡❝❤❛❞❛ ❡ s✐♠♣❧❡s♠❡♥t❡
❝♦♥❡①❛✳ ❈♦♥s✐❞❡r❡ (Mn, g) ❛ ✈❛r✐❡❞❛❞❡ ❡stát✐❝❛ ❛ss♦❝✐❛❞❛ ❛ N✳ ❙❡ 3 ≤ n ≤ 7✱ ∂M t❡♠
♠❛✐s ❞❡ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❝♦♥❡①❛ ❡ Rg(p)≤n(n−1)✱ ♣❛r❛ t♦❞♦p∈∂M✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛
❤✐♣❡r❢í❝✐❡ q✉❛s❡✲❊✐♥st❡✐♥✱ ❤♦♠ó❧♦❣❛ à ∂M ❡ ♣r♦♣r✐❛♠❡♥t❡ ♠❡r❣✉❧❤❛❞❛ ❡♠ M✳
❙❛❧✐❡♥t❛♠♦s q✉❡ ♦ ❚❡♦r❡♠❛ ❛❝✐♠❛ ❣❡♥❡r❛❧✐③❛ ✜❡❧♠❡♥t❡ ♣❛r❛ ❞✐♠❡♥sõ❡s 3 ≤ n ≤ 7 ♦
t❡♦r❡♠❛ q✉❡ ♣r♦♣♦♠♦s ❛ ❣❡♥❡r❛❧✐③❛r ❞❡ ❬✽✷❪✳ ◆♦ ❝❛s♦ ♣r♦♣♦st♦ ♣♦r ❙❡s❤❛❞r✐✱ ❛ ❞✐♠❡♥sã♦
3 ♣r♦♣õ❡ ❛ ❝❧❛ss✐✜❝❛çã♦ ❝♦♠♣❧❡t❛ ❞❛ ✈❛r✐❡❞❛❞❡ q✉❛s✐✲❊✐♥st❡✐♥ ♦❜t✐❞❛ ❡✱ ♣♦r ❝♦♥s❡❣✉✐♥t❡✱
❈❛♣ít✉❧♦
1
Pr❡❧✐♠✐♥❛r❡s
✶✳✶ ◆♦t❛çõ❡s✱ t❡r♠✐♥♦❧♦❣✐❛ ❡ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s
❆♦ ❝✉rs♦ ❞❡st❡ tr❛❜❛❧❤♦✱ s❛❧✈♦ ♠❡♥❝✐♦♥❛❞♦ ❞❡ ♦✉tr❛ ❢♦r♠❛✱ (Mn, g) ❞❡♥♦t❛rá ✉♠❛
✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐♥❛ ❝♦♠♣❛❝t❛ ❡ s✉❛✈❡ ❞❡ ❞✐♠❡♥sã♦ n ❡ ♠étr✐❝❛ s✉❛✈❡ g✳ D ❞❡♥♦t❛rá ❛
❝♦♥❡①ã♦ ❞❡ ▲❡✈✐ ❈✐✈✐t❛ ❛ss♦❝✐❛❞❛ à ✈❛r✐❡❞❛❞❡ (M, g)✳ ❆ ✉♠ ♣♦♥t♦ x∈M✱ ❞❡♥♦t❛♠♦s ♣♦r TxM ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❞❡ M ❡♠ x ❡ X(M) ♦ ❡s♣❛ç♦ ❞❡ t♦❞♦s ❝❛♠♣♦s ✈❡t♦r✐❛✐s s✉❛✈❡s
❡♠ M✳ ❯t✐❧✐③❛r❡♠♦s ❡♠ ♥♦ss♦ t❡①t♦ ❛ ❝♦♥s❛❣r❛❞❛ ♥♦t❛çã♦ ❞❡ ❊✐♥st❡✐♥✱ s❡❣✉♥❞♦ ❛ q✉❛❧
♦♠✐t✐♠♦s ♦ sí♠❜♦❧♦ ❞❡ s♦♠❛tór✐♦ ❛♦ ✐♥t❡♣r❡t❛r ✐♥❞í❝❡s r❡♣❡t✐❞♦s ♥✉♠ ♠❡s♠♦ t❡r♠♦ ❝♦♠♦ ✐♥❞✐❝❛❞♦r ❞❡st❡ s♦♠❛tór✐♦✳
◆❛s ♣ró①✐♠❛s s✉❜s❡çõ❡s✱ ❡st❛❜❡❧❡❝❡r❡♠♦s ❛❧❣✉♠❛s ♥♦t❛çõ❡s ❡ r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ❞❡ ●❡♦♠❡tr✐❛ ❘✐❡♠❛♥♥✐❛♥❛✱ ✐♠♣♦rt❛♥t❡s ❛♦ ❝✉rs♦ ❞❡ ♥♦ss♦ t❡①t♦✳ ❆ ♠❛✐♦r✐❛ ❞❛s ♥♦t❛çõ❡s ❡ ❞❡♠♦♥str❛çõ❡s ❞♦s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❛❧❣✉♥s ❝♦♥s❛❣r❛❞♦s ❧✐✈r♦s ❞❛ ♠❛tér✐❛ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✷✾❪✱ ❬✺✾❪ ❡❬✼✷❪✳
✶✳✶✳✶ ❆ ❣❡♦♠❡tr✐❛ ✐♥trí♥s❡❝❛
❉❛❞♦s X, Y, Z, W ∈ X(M) ❝❛♠♣♦s s✉❛✈❡s✱ ❞❡✜♥✐♠♦s ♦ (1,3) ❚❡♥s♦r ❈✉r✈❛t✉r❛ ❘✐❡✲
♠❛♥♥✐❛♥❛✱ ❞❡♥♦t❛♥❞♦✲♦ ♣♦r R✱ ❝♦♠♦
R(X, Y)Z =DXDYZ−DYDXZ −D[X,Y]Z,
❡ ❛♦ t❡♥s♦r (0,4)❞❡ ❝✉r✈❛t✉r❛ t❛♠❜é♠ ❞❡♥♦t❛❞♦ ♣♦r R ♣♦r
✶✳✶ ◆♦t❛çõ❡s✱ t❡r♠✐♥♦❧♦❣✐❛ ❡ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ✶✵
❉❛❞♦sx∈M ❡π ⊂TxM ✉♠ ♣❧❛♥♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✱ ❞❡✜♥✐♠♦s ❛ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❞❡
π ♣♦r
K(π) = R(X, Y, X, Y)
|X|2|Y|2− hX, Yi2,
♦♥❞❡ {X, Y} é ✉♠❛ ❜❛s❡ ❞❡ π✳ ❆ ❞❡✜♥✐çã♦ ❞❡ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡ ❞❛ ❜❛s❡
{X, Y} ❡s❝♦❧❤✐❞❛✳
❋✐①❛❞♦ x∈ M✱ ❝♦♥s✐❞❡r❡ {e1, . . . , en} ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ TxM✳ ❆ ❝✉r✈❛t✉r❛ ❞❡
❘✐❝❝✐ ❞❡ (M, g)é ✉♠ t❡♥s♦r ❞♦ t✐♣♦ (0,2)✱ ❞❡✜♥✐❞♦ ♣♦r
Ricg(X, Y) = n
X
k=1
R(X, ek, Y, ek),
❡ ❛ ❝✉r✈❛t✉r❛ ❡s❝❛❧❛r é ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ ❡♠ M ❞❡✜♥✐❞❛ ♣♦r
Rg(x) = n
X
k=1
Ricg(ek, ek)(x).
P♦r ✜♠✱ ♦ t❡♥s♦r ❞❡ ❘✐❝❝✐ s❡♠ tr❛ç♦ ❞❡ (M, g)é ❞❡✜♥✐❞♦ ♣♦r
◦
Ricg(X, Y) =Ricg(X, Y)−
1
nRg·g(X, Y).
✶✳✶✳✷ ❆ ❣❡♦♠❡tr✐❛ ❡①trí♥s❡❝❛
❈♦♥s✐❞❡r❡Σn−1 ⊂M ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❝♦♠♣❧❡t❛✱ ✐♠❡rs❛ ❡ ✷✲s✐❞❡❞ ❡♠(Mn, g)✳ P❛r❛
❝❛❞❛x∈Σ✱ ❝♦♥s✐❞❡r❡TxΣ⊂TxM ♦ ❡s♣❛ç♦ t❛♥❣❡♥t❡ ❛ x❡ν ∈X(Σ)⊥⊂X(M)✉♠ ❝❛♠♣♦
♥♦r♠❛❧ ❡①t❡r✐♦r ✉♥✐tár✐♦ à Σ✳ ❖ ♦♣❡r❛❞♦r ❞❡ ❢♦r♠❛ ❞❡ Σ é ❞❡✜♥✐❞♦ ❝♦♠♦
S(X) :=DXν,
♦♥❞❡ X ∈ X(Σ)✳ ❆ ❙❡❣✉♥❞❛ ❋♦r♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ Σ é ✉♠❛ ❢♦r♠❛ ❜✐❧✐♥❡❛r ❡ s✐♠étr✐❝❛
❡♠ X(Σ)✱ ❞❡✜♥✐❞❛ ❝♦♠♦
II(X, Y) := g(DXν, Y).
❆ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❞❡ Σ♥✉♠ ♣♦♥t♦ x∈Σ é ❞❡♥♦t❛❞❛ ♣♦rH(x) ❡ ❞❡✜♥✐❞❛ ❝♦♠♦ ♦ tr❛ç♦
✶✳✶ ◆♦t❛çõ❡s✱ t❡r♠✐♥♦❧♦❣✐❛ ❡ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ✶✶
H(x) := tr❛ç♦{Y(x)→DYν(x)}
=
n−1 X
i=1
II(ei, ei),
♦♥❞❡ {e1, . . . , en−1} ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ TxΣ✳
❈♦♥s✐❞❡r❡ ❝❛♠♣♦s X, Y, Z, W ∈X(Σ)✳ ❊①✐st❡ ✉♠❛ ✐♠♣♦rt❛♥t❡ r❡❧❛çã♦ ❡♥tr❡ ❛ ❝✉r✈❛✲
t✉r❛ ❛♠❜✐❡♥t❡ ❡ ❛ ❝✉r✈❛t✉r❛ ✐♥trí♥s❡❝❛ ❞❛ ❤✐♣❡r❢í❝✐❡ Σ ❞❛❞❛ ♣❡❧❛ ❊q✉❛çã♦ ❞❡ ●❛✉ss
R(X, Y, Z, W) = RΣ(X, Y, Z, W)−II(X, W)II(Y, Z) +II(X, Z)II(Y, W),
♦♥❞❡ R ❡ RΣ ❞❡♥♦t❛♠ ❛ ❝✉r✈❛t✉r❛ ❞❡ M ❡ Σ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❚♦♠❛♥❞♦ ❞✉❛s ✈❡③❡s ♦
tr❛ç♦ ❞❛ ❡q✉❛çã♦ ❛❝✐♠❛✱ t❡♠♦s q✉❡
Rg −2Ricg(ν, ν) =RΣg −H2+|II|2,
♦♥❞❡ Ric(ν, ν) é ♦ t❡♥s♦r ❞❡ ❘✐❝❝✐ ❞❡ M ❡✈❛❧✉❛❞♦ ♥❛ ❞✐r❡çã♦ ♥♦r♠❛❧ à Σ ❡ |II| ❞❡♥♦t❛ ❛
♥♦r♠❛ ❞❛ s❡❣✉♥❞❛ ❢♦r♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ Σ✳
✶✳✶✳✸ ■❞❡♥t✐❞❛❞❡s ❘✐❡♠❛♥♥✐❛♥❛s
❈♦♥s✐❞❡r❡ ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ (Mn, g) ❡ f : M −→ R ✉♠❛ ❢✉♥çã♦ s✉❛✈❡✳
❉❡♥♦♠✐♥❛r❡♠♦s ♣♦r ∇f ❛♦ ❝❛♠♣♦ q✉❡ s❛t✐s❢❛③
g(∇f, X) =df(X),
♣❛r❛ t♦❞♦ X ∈X(M)✳
❉❡✜♥✐♠♦s ♦ ❍❡ss✐❛♥♦ ❞❡f✱ s❡❣✉♥❞♦ ❛ ♠étr✐❝❛ g✱ ❝♦♠♦ ♦ s❡❣✉✐♥t❡ 2✲t❡♥s♦r
D2f(X, Y) =DXDYf−DDXYf, ♦♥❞❡ X, Y ∈X(M)✳
❉❛❞♦x∈M✱ ♦ ▲❛♣❧❛❝✐❛♥♦ ❞❡ f é ❞❡✜♥✐❞♦ ♣♦r
∆f(x) =
n
X
i=1
D2f(ei, ei),
♦♥❞❡ {e1, . . . , en}é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ TxM✳
✶✳✶ ◆♦t❛çõ❡s✱ t❡r♠✐♥♦❧♦❣✐❛ ❡ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ✶✷
t❡①t♦✱ ✐♥❞✐❝❛♥❞♦ r❡❢❡rê♥❝✐❛s ♣❛r❛ s✉❛s ❞❡♠♦♥str❛çõ❡s✳
❚❡♦r❡♠❛ ✶✳✶ ✭■❞❡♥t✐❞❛❞❡s ❞❡ ❇✐❛❝❤✐✮✳ ❖ t❡♥s♦r ❈✉r✈❛t✉r❛ ❘✐❡♠❛♥♥✐❛♥❛ R s❛t✐s❢❛③ ❛s
s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♣❡r♠✉t❛çã♦ ❝í❝❧✐❝❛s✿
✭✐✮ ❆ ♣r✐♠❡✐r❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❇✐❛♥❝❤✐✿
R(X, Y)Z +R(Z, X)Y +R(Y, Z)X = 0,
♦♥❞❡ X, Y, Z ∈X(M)❀
✭✐✐✮ ❆ ❙❡❣✉♥❞❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❇✐❛♥❝❤✐✿
(DXR)(Y, Z)W + (DZR)(X, Y)W + (DYR)(Z, X)W = 0,
♦♥❞❡ X, Y, Z, W ∈X(M).
❚♦♠❛♥❞♦ ♦ tr❛ç♦ ❞❡st❛ ✐❞❡♥t✐❞❛❞❡✱ ❝❤❡❣❛♠♦s ♥❛ ❙❡❣✉♥❞❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❇✐❛♥❝❤✐ ❝♦♥tr❛í❞❛✿
divRicg =
1 2∇Rg.
❉❡♠♦♥str❛çã♦✳ ❱❡❥❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ♣á❣✐♥❛ ✸✸ ❞❡ ❬✼✷❪✳
❚❡♦r❡♠❛ ✶✳✷ ✭❋ór♠✉❧❛ ❞❡ ❇♦❝❤♥❡r✲❲❡✐t③❡♥❜ö❝❦✮✳ ❈♦♥s✐❞❡r❡ (M, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐✲
❡♠❛♥♥✐❛♥❛✳ ❉❛❞❛ ✉♠ ❢✉♥çã♦ s✉❛✈❡ u:M −→R✱ t❡♠♦s q✉❡
1 2∆|du|
2 =
|D2u|2 +h∇∆u,∇ui+Ricg(∇u,∇u).
❉❡♠♦♥str❛çã♦✳ ❱❡❥❛✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❈❛♣ít✉❧♦ ✼ ❞❡ ❬✼✷❪✳
P❛r❛ ✜♥❛❧✐③❛r ❡st❛ s❡çã♦✱ ❧❡♠❜r❛r❡♠♦s ❞❛ ❢❛♠♦s❛ ❋ór♠✉❧❛ ❞❡ ❘❡✐❧❧② ❡ ❛❧❣✉♠❛s ❞❡ s✉❛s ❛♣❧✐❝❛çõ❡s ❡♠ r❡s✉❧t❛❞♦s ❞❡ r✐❣✐❞❡③✳
❚❡♦r❡♠❛ ✶✳✸ ✭❋ór♠✉❧❛ ❞❡ ❘❡✐❧❧②✱ ❬✼✽❪✮✳ ❈♦♥s✐❞❡r❡ (M,h,i) ✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛
❝♦♠ ❢r♦♥t❡✐r❛ Σ ❡ u:M −→R ✉♠❛ ❢✉♥çã♦ s✉❛✈❡✳ ❊♥tã♦✱ t❡♠♦s ❛ ■❞❡♥t✐❞❛❞❡
1 2
Z
M
(∆u)2− |D2u|2
dV ol = 1 2
Z
M
Ric(∇u,∇u)dV +
Z
Σ
∆Σu· ∂u
∂νdS
+1 2
Z
Σ H
∂u ∂ν
2
dS+ 1 2
Z
Σh
✶✳✷ ❱❛r✐❡❞❛❞❡s ❡stát✐❝❛s ✶✸
♦♥❞❡ ν é ♦ ❝❛♠♣♦ ♥♦r♠❛❧ ❡①t❡r✐♦r✱ II(·,·) ❡ H sã♦ ❛ ❙❡❣✉♥❞❛ ❋♦r♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❡ ❛
❝✉r✈❛t✉r❛ ♠é❞✐❛ ❞❡ Σ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆❧é♠ ❞♦ ♠❛✐s✱ ♦ ✐♥❞í❝❡ Σ s♦❜ ∇ ❡ ∆ ✐♥❞✐❝❛ q✉❡
❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ❡st❡s ❡♥t❡s ✐♥tr✐s❡❝❛♠❡♥t❡ ❛ Σ✳
❚❡♦r❡♠❛ ✶✳✹ ✭❘❡✐❧❧②✱ ❬✼✽❪✮✳ ❈♦♥s✐❞❡r❡(M, g)✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❛❝t❛ ❝♦♠
❢r♦♥t❡✐r❛ ♥ã♦ ✈❛③✐❛ ∂M✳ ❆ss✉♠❛ q✉❡ Ricg ≥ (n−1)g ❡ q✉❡ ❛ ❝✉r✈❛t✉r❛ ♠é❞✐❛ ❞❡ ∂M
é ♥ã♦✲♥❡❣❛t✐✈❛✭∂M é ♠❡❛♥ ❝♦♥✈❡①✮✳ ❊♥tã♦ ♦ ♣r✐♠❡✐r♦ ❛✉t♦✈❛❧♦r λ1 ❞❡ −∆ s❛t✐s❢❛③ ❛
❞❡s✐❣✉❛❧❞❛❞❡ λ1 ≥ n✳ ❆❧é♠ ❞♦ ♠❛✐s✱ λ1 = n s❡✱ ❡ s♦♠❡♥t❡ s❡✱ M é ✐s♦♠étr✐❝❛ ❛♦
❤❡♠✐s❢ér✐♦ ❢❡❝❤❛❞♦ ❞❛ ❡s❢❡r❛ ❡✉❝❧✐❞✐❛♥❛ Sn✳
❚❡♦r❡♠❛ ✶✳✺ ✭❘❡✐❧❧②✱ ❬✼✾❪✮✳ ❈♦♥s✐❞❡r❡ (M, g) ✉♠❛ ✈❛r✐❡❞❛❞❡ ♥✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♠♣❛❝t❛✱
♦r✐❡♥t❛❞❛✱ ❝♦♥❡①❛ ❝♦♠ ❢r♦♥t❡✐r❛ ♥ã♦ ✈❛③✐❛✳ ❙✉♣♦♥❤❛ q✉❡✱ ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ c 6= 0✱
M ❛❞♠✐t❡ ✉♠❛ ❢✉♥çã♦ ♥ã♦ ❝♦♥st❛♥t❡f :M −→R t❛❧ q✉❡
✭❛✮ D2f =−cf g❀
✭❜✮ ❢ é ❝♦♥st❛♥t❡ s♦❜r❡ ∂M✳
❊♥tã♦✱ ❛ ♠étr✐❝❛ g ❡♠ M t❡♠ ❝✉r✈❛t✉r❛ s❡❝❝✐♦♥❛❧ ❝♦♥st❛♥t❡ c✳
✶✳✷ ❱❛r✐❡❞❛❞❡s ❡stát✐❝❛s
❈♦♠♦ ❥á ❝♦♠❡♥t❛♠♦s ♥❛ ✐♥tr♦❞✉çã♦✱ ♦ ❡st✉❞♦ ❞❡ ✈❛r✐❡❞❛❞❡s ❡stát✐❝❛s ♣❛r❛♠❡tr❛ ✉♠ ✐♠♣♦rt❛♥t❡ ❝♦♠♣♦♥❡♥t❡ ♥❛ ❛✈❛❧✐❛çã♦ ❞❛s s♦❧✉çõ❡s ❞❛s ❡q✉❛çõ❡s ❞❡ ❊✐♥st❡✐♥ ♥♦ ✈á❝✉♦✳ ❉❡ ❢❛t♦✱ ❞❛❞❛ ✉♠❛ (Nn+1, h) ✈❛r✐❡❞❛❞❡ ▲♦r❡♥t③ q✉❡ é ✉♠❛ s♦❧✉çã♦ ❡stát✐❝❛ ❞❛ ❡q✉❛çã♦
❞❡ ❊✐♥st❡✐♥ ♥♦ ✈á❝✉♦✱ ♣♦❞❡✲s❡ ❝❛r❛❝t❡r✐③❛r N ❝♦♠♦ ✉♠ ❡s♣❛ç♦ t❡♠♣♦ q✉❡ é ✉♠ ♣r♦❞✉t♦
✇❛r♣❡❞ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ t❡♠♣♦r❛❧ ♣♦r ✉♠❛ ✈❛r✐❡❞❛❞❡ ❡stát✐❝❛✳ ◆❡st❛ s❡çã♦✱ r❡❧❡♠❜r❛♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❛r✐❡❞❛❞❡ ❡stát✐❝❛ ❡ ❡st❛❜❡❧❡❝❡r❡♠♦s ♣r♦♣r✐❡❞❛❞❡s ♥♦tá✈❡✐s ❞❛s ♠❡s♠❛s✱ ❜❡♠ ❝♦♠♦ ❛❧❣✉♥s ❝♦♥❤❡❝✐❞♦s r❡s✉❧t❛❞♦s ❞❡ r✐❣✐❞❡③ q✉❡ ❝❛♠✐♥❤❛♠ ♥❛ ❞✐r❡çã♦ ❞❛ ✈❡r❛❝✐❞❛❞❡ ❞❛ ◆♦✲❍❛✐r ❈♦♥❥❡❝t✉r❡✳
❉❡✜♥✐çã♦ ✶✳✶✳ ❈♦♥s✐❞❡r❡(Mn, g)✉♠❛ ✈❛r✐❡❞❛❞❡ ❘✐❡♠❛♥♥✐❛♥❛ ❝♦♠♣❧❡t❛ ❝♦♠ ❜♦r❞♦(∂M 6=
∅)✳ ❉✐③❡♠♦s q✉❡(Mn, g)é ✉♠❛ ✈❛r✐❡❞❛❞❡ ❡stát✐❝❛ s❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ s✉❛✈❡ ♥ã♦ ♥❡❣❛t✐✈❛
f :M −→R ❡ ✉♠❛ ❝♦♥st❛♥t❡ Λ✱ t❛❧ q✉❡ ∂M =f−1(0)✱ ∆f =−Λf ❡
f Ricg−(Λf)g−D2f = 0, ✭✶✳✶✮
